Dissolve the dry yeast in lukewarm milk. Add the flour, softened butter, salt, sugar, and egg to a bowl, add the yeast milk mixture, and combine everything with a hand mixer with dough hooks until smooth. Cover the bowl and let the dough rise for about 1 hour. After 1 hour, knead the dough on a floured work surface and shape it into a 16 x 12-inch loaf. Place the brioche loaf on a baking tray lined with parchment paper and let the dough rise for another 10 minutes. Press 3 ramekins (⌀ 4 inch), evenly spaced out, into the dough at the top of the dough to create indents. Do the same at the bottom of the dough and leave the ramekins in the dough.
Peel the onions and use a melon baller to scoop out the insides. Line a 12 x 4-inch baking tray with parchment paper and arrange the American cheese slices in 2 rows on one of the baking trays.
Season the ground beef with salt, pepper, and paprika and spread the mixture over the cheese slices. Place a piece of parchment paper on top of the ground beef and press the bottom of another 12 x 4-inch baking tray into the mixture to flatten it down. Flip the ground beef and cheese rectangle out onto the countertop, remove the parchment paper, and use a pizza cutter to cut the rectangle into 3 horizontal strips.
Roll each ground beef and cheese strip up and place one inside each hollowed-out onion.
Wrap 6 slices of bacon around each stuffed onion.
Place the bacon cheeseburger onions in the indents in the brioche dough and brush BBQ sauce over the bacon.
Place mozzarella pearls and crispy fried onions in the first ramekin. Place cheddar cheese cubes and chili pepper slices in the second ramekin. Place Swiss cheese cubes and diced pickles in the third ramekin.
Brush the egg wash over the dough and sprinkle sesame seeds on top. Transfer the baking tray to a preheated oven and cook at 350°F for 35 minutes.
Carefully remove the hot ramekin dishes from the brioche loaf and stir the cheese sauces in each dish. Cut the baked brioche loaf in half horizontally and place the side with the holes on top of the side with the BBQ bacon cheeseburger onions. Pour a cheese sauce over each BBQ bacon cheeseburger onion and garnish the brioche loaf with diced tomatoes and shredded lettuce.