doughnut post-mortem: mis-shapen, overcooked

Thanks for the input, Lori and Nancy. These are great comments. (I found a Youtube video this morning of Joanne Chang making these doughnuts with Martha Stewart and yup, right away noticed the deep pot. I had forgotten, too, that I had recently purchased the requisite thermometer. Next time will aim for a less-complicated recipe, appropriate tools and timing. In the process I did wonder why the recipe didn't specify a few details--a greased bowl, punching down after the first rise. Anyway, I made a good number of mistakes (did I mention the first round of crème pat coming out like a hockey puck?--went back to my old favorite recipe--which always works for me). Live and learn!
Happy Chanukah and New Year!

Creamtea - my reactions and advice are similar to those of Lori T.
Key to making successful deep fried food, doughnuts included, is to have plenty of oil, in a deep container, get it to and maintain right temp for cooking. (I have had great success with a deep soup pot or Dutch oven. Vertical structure gets you better depth with less oil. But, in any case, bite the bullet and be generous with the oil. I would go for the minimum 3" or more.
1) Get the oil to the right 375F temp. (Do you have a so-called candy thermometer? Long, and can be used for jam, candy, frying. Worth it.)
2) Use a deep pan. Avoid a wide frying pan because you then you need even more oil to get minimum depth, but also get more surface area to make splatters.
3) Let the raw items have plenty of space and don't fry too many at once or their combined presence will reduce the oil temp too much.
When all 3 conditions are present, the doughnuts will quickly form a crisp outer seal and cook the insides faster.
When oil temp is not hot enough to start and/or pan too crowded and oil temp drops, the batter doesn't form the outer coating, the dough absorbs more oil AND cooks more slowly.
That's probably why your doughnuts were an unlikely combination of greasy, burnt and not quite cooked through.
May also have contributed to the collapse (versus puff up) of the yeasty dough structure.
As for the recipe...I have made brioche for holidays, and they're wonderful, but not easy in their native baked state. And, I imagine, even harder to fry (with managing the butter absorption, waiting time while making the doughnuts in sequence.)
I would go for an easier, more basic dough to first master the technique, like one of the two attached - one from allrecipes site, one from Mark Bittman in nyt (each used & liked by hundreds of home cooks). Then, if you want, go back to the brioche type doughnut.
https://www.allrecipes.com/recipe/72964/super-easy-doughnuts/
https://cooking.nytimes.com/recipes/1017060-doughnuts
Hag sameach, 4 nights down, 4 to go.
Nancy

Hey, Creamtea! Happy Chanuhka, and Happy New Year. So there are a couple places I can see that might have led to your troubles. Is it possible that your initial dough needed just a little more flour and kneading? Brioche type doughs can be fiddly to deal with because it's hard to judge this sometimes. It should be kneaded for at least 10 minutes or longer, to clear the bowl and make a cohesive dough that is just slightly sticky to touch. If it won't somewhat hold shape in the rising bowl, it's not going to improve in the resting time, when the flour finishes absorbing the rest of the liquid. That would leave you with floppy cut shapes, which can't hold their shape because there isn't enough gluten development or flour to do the job. Then, when the donuts are almost fully risen the second time, I like to remove the plastic wrap for the last 5-10 minutes to let the dough dry slightly and form sort of a skin. I find that helps it crisp, and avoid absorbing quite so much of the oil. If you can, keep the risen donuts somewhat cool while they wait on their turn in the fryer. If the butter or shortening in that dough starts getting too warm- you end up with sticky dough that will just absorb more oil.
If you need to, and have the room, let those waiting get a chill on in the fridge for a few moments before they hit the oil. You don't want them cold- but you don't want them too warm. Finally, your frying method may have been a problem as well. The depth of the oil is really critical for donuts- as they really do need to swim and float in the hot stuff. You just have to concede to having a good three inches of oil, and not be tempted to use less. Also you have to resist the temptation to fry more than a few at a time, so the oil temperature doesn't drop and they are not crowded. Oil temperature is also very crucial- and it's worth it to invest in a deep frying thermometer so you can monitor the heat and adjust as needed. My favorite deep fry pan is actually an tie between a regular cast iron dutch oven and an enamel coated cast iron version. I pick depending on how much I'm frying, actually. Cast iron is wonderful for deep frying- beaten only by a really good purpose made deep fryer that lets you set temperature. The only other thing I can suggest is to make donuts with this recipe again, a few times maybe- to get the feel for how it should look and feel to work. Maybe not so good on the waistline, but you can always share- nobody complains about getting a donut. Then when Chanuhka comes again, you will be a dap hand at making them and the hit of the feast.