Here is polenta cooked as above.

I found that if you refrigerate for several hours or peferably overnight, heat a non-stick pan with equal quantities of olive oil/butter until hot, then add the sliced polenta after you have dipped in all purpose flour and shaken off the excess, then cook on first side until desired level of brown and crunchy is acheived, then the other side the same, you will have success EVERY time and the best polenta ever!

Liked many of the ideas from others.
If y'all said this & I missed it, please forgive.
But in addition to the cooling of the polenta, and the oil tricks, I also grill it...inside under the broiler, or on an outside grill if the pieces are solid enough and you have one of those baskets to prevent it from falling into the fire.
Also, if you like & appropriate to the meal, add a bit of grated parmesan for a nice crust.
Do grill marks count as "brown polenta"?
You decide.

You can fry polenta after keeping it in the fridge for about 20 to 30 minutes after cooking. To cool quickly, spread a layer of about 3/4 to inch thick on a plate or small baking sheet. I fry mine in butter with a bit of olive oil to keep the butter from burning.
The polenta will hold together better if a beaten egg is added to the cooled polenta and/or 2 to 3 tablespoons of butter are added. Lard works really great as it solidifies well and your polenta will hold together even better as well as self baste.
I use a nonstick pan and it works great. Non stick pans are primarily aluminum and heat well and thoroughly. Regular pans, good stainless steel, work well too but there is a tendency for the polenta to stick.
Unfortunately, if you use coconut oil, the frying process takes much longer. On the plus side, coconut oil can take the high heat without burning. : )

See ChefDaddy's answer ! You *aren't* trying to brown freshly made polenta, are you, 'cuz that ain't gunna happen.

Make a day ahead (at least), make more than you need, and enjoy day 1 as polenta. Then pour the rest into a sheet/loaf/flat pan and refrigerate at least over night. Cut into squares, diamonds, circles, whatever, flour lightly and brown in oil, butter or whaterver. Works like a charm. Or just make enough for the chilled, cut and fried, if you don't want the initial polenta. But to fry it, it must, must be firmed up and chilled for a good long time. No broiler, the shallow fry is the way to go.

If were talking about polenta that has been poured,cooled, set and cut into shapes. Add butter AND olive oil to a pan and get nice and hot before you add polenta. It will brown up nicely.

betteirene is right about reducing the amount of liquid, which ties into what mrslarkin says about using thicker polenta.

Just to make sure, you are cooking the polenta the day before frying, right? I think that's important. Cook the day before and put it in a loaf pan and leave it in the fridge overnight, then slice it before you fry it.

The kind of pan you fry in also makes a difference. non-stick may be convenient, but I don't think it gives you a nice a brown as a non-non-stick.

I also find that softer polenta doesn't crisp up as nicely. So when I intend to pan fry, (and I only ever pan fry polenta) I make a batch of thick polenta (just use less water.)

Working at a burger joint at the dawn of the fast food era and at a truck stop might not have been the most glamorous of jobs, but I sure did learn a lot of nifty little tricks from fry cooks.

First, you have to get as much moisture off the surface of the food as possible so that the hot oil doesn't waste its time boiling off the moisture (and creating steam) before it can begin to make the food brown and crisp.. You can do this by dredging with something dry, like flour or bread crumbs, or by letting the food air dry, or by a combination of both.

Second, you can add something to the polenta itself, like a bit of sugar or some grated cheese (parmesan, yum), to help things along.

Third, and the most bestest trick ever for better frying (which actually is Rule #1), add a small amount of old (previously used and cooled) oil to the new oil. Try this experiment: In one small skillet, heat a half cup of oil with a couple of tablespoons of used oil; in another small skillet, heat a half cup of fresh, clean, never-used oil, and fry equal-sized slices of potato or polenta in each skillet. I'll bet $20 that the one with a bit of used oil browns quicker, crisper and more evenly than the fresh one. A well-stocked kitchen should always have a small container of used oil, like the crock of bacon grease our grandmothers used to keep on the stove. It doesn't matter what kind of oil is used for frying, by the way--even olive oil needs a little nudge when it comes to crisping things up.

I don't think broiling is the answer. Yes, the heat is higher, which is a decent crisping and browning aid in and of itself, but I think your results would be spotty.

I always broil it and it turns out really nice and crispy on the outside, but still smooth and soft on the inside. Cover your broiling pan with aluminum foil and brush with olive oil. Preheat the broiler to about 400F (425 if your oven tends to be on the cool side). Lay the slices of polenta out on the foil. Brush the polenta with olive oil. Broil about 15min, turn them over, brush with more olive oil, and broil about 10 more min. You'll know they're done when the slices are browned and the edges are beginning to be dark brown.

Thanks for the suggestions. I am going to dust it with flour and try it that way. I am also thinking something involving the broiler. Open to all advice!

I'm going to reply since you don't have any answers yet, but someone else may have more experience with them. The ones I used to do were roasted in a 500F oven and did get golden but did not have the level of crispiness you are seeking. I want to make again soon.
http://books.google.com/books?id=H3tz-11DlXkC&pg=PA87&lpg=PA87&dq=roasted+polenta+San+Francisco+Chronicle+cookbook&source=bl&ots=j8v-MBGna1&sig=LtJ0ee1LvJD74Sc-dQVty1d-Mtw&hl=en&ei=d3A3Td2DDpD6sAOx99jbAw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBMQ6AEwAA#v=onepage&q&f=false

I notice that Joanne Weir nearly deep fries them. She dusts with flour and fries in 1/2 inch of olive oil heated to 375F. This sounds like it would get pretty crispy to me.
http://www.foodnetwork.com/recipes/sara-moulton/crispy-polenta-cakes-with-wild-mushrooms-ragout-recipe/index.html