Melt 1 tablespoon butter in a large skillet over medium-low heat. Add onions and stir occasionally until onions are a deep caramelized color, about 35-40 minutes.
Add garlic and cook 1 minute until fragrant. Pour balsamic vinegar into pan and deglaze, scraping up any browned bits. Place caramelized onions in a bowl and set aside.
Melt remaining 1 tablespoon butter in same skillet over medium-high heat. Add peppers to skillet and sauté 4-6 minutes until peppers are tender. Toss peppers into caramelized onion mixture and lightly season with salt and pepper.
Divide caramelized onion/pepper mixture in half. Place one of the portions on a cutting board and finely dice. Reserve other half on a small sheet of foil for topping on burgers later.
Place diced portion of caramelized onion/pepper mixture in a large bowl and add ground chuck, salt, pepper, paprika, and onion powder. Gently knead until meat is just combined then divide into 4 portions. Form each portion into a burger patty.
Preheat an outdoor grill to 400F or if cooking indoors, grease a large cast iron grill pan and heat over medium-high heat. Grill burgers flipping once until burgers reach your desired degree of doneness (see blog post for more info on this) Add cheese slices during last 1-2 minutes of cooking to melt on top of burgers.
Place reserved caramelized onion/peppers mixture on foil on grill and cook 2-3 minutes, tossing frequently until heated through.
Arrange lettuce leaves on bottom half of toasted buns. Top with burger patty, tomato slices, and reserved caramelized onion/peppers mixture. Top with top half of buns, serve burgers immediately, and enjoy!