Peel and thinly slice 2 onions. Heat a tea spoon of oil in a pan to medium, then add onion slices and lightly fry until the onions are translucent. Add about 1/3 of beer and let it simmer until the beer has cooked down. Add brown sugar and balsamic vinegar, stir together, then add another 1/3 of beer and cook for 5-10 minutes, until the onions are soft, lightly brown and in a creamy sauce. Season with salt and pepper to taste. Remove from stove and set aside.
While the onions are cooking, prepare Beer Batter Crispy Onion Rings: mix together about 2-3 tbs of flour and about 1/3 of beer. Season the batter with salt and pepper. Peel an onion and cut into slices. I made thin slices, but you can make them wider. Add onion slices to the beer batter and stir in gently, making sure the onion slices are covered in batter. Heat up oil for frying, then add onion slices one by one. Make sure the onions have enough room to float around in oil. Fry for about a minute on each side, or until the onion rings and golden-brown and crispy, then remove from pan and set on a plate, covered with a paper towel.
Heat up the grill pan and rub with a bit of oil. Place burger patties on grill and cook about 4 minutes until golden brown, then turn around and grill another couple of minutes on the other side. Remove from pan and set aside. Cut buns in half and place them on the grill pan to grill for a couple of minutes. Remove from pan.
In a small bowl, mix together mayonnaise and mustard. Wash and dry tomato, then cut into slices.
Start assembling your burgers. Take the bottom bun slices and spread with the mayonnaise mix. Add burger patty on top and a tomato slice, then top with beer braised onions. Add about 1 ts of barbecue sauce (I used store-bought), then top with a few crispy onion rings and some lettuce. Finish the burger with the top bun slice and stick through with a bamboo stick or knife. Serve.