Preheat broiler.
Place a large pan over medium-high heat with 2 tablespoons EVOO, about 2 turns of the pan. Add the onions and season with salt and freshly ground black pepper. Place a lid or piece of aluminum foil over the pan and cook the onions, stirring occasionally, until they start to soften, 2-3 minutes. Remove the lid or foil from the pan and continue cooking until fully softened, another 4-5 minutes.
Add the tomato paste to the pan and cook, stirring occasionally, until it smells sweet and has caramelized slightly, 2-3 minutes. Add the wine and scrape the pan with a wooden spoon to remove any bits that may be stuck to the bottom. Cook until the wine has almost completely evaporated, 3-4 minutes. Remove the onions from the heat, cover them again with the lid or aluminum foil and reserve until ready to serve.
Place a large skillet over medium-high heat with three turns of the pan of EVOO, about 3 tablespoons. In a large mixing bowl, combine the ground sirloin, pepperoni, parsley, oregano and red pepper flakes. Season the meat with salt and freshly ground black pepper, then flatten the meat down in the bowl and score it into four using the side of your hand. Form a patty with each of the four portions of the meat and place them into the hot skillet. Cook until desired doneness, 4-5 minutes per side for medium.
While the burgers are cooking, place a small pot over medium heat with the garlic. Melt the butter then remove the pot from the heat to allow the garlic flavor to infuse the butter. While the butter is melting, place the bread slices on a baking sheet and toast them under the broiler until golden brown on each side. Brush the toasts with the garlic butter and sprinkle the grated cheese over each of them.
Place four toasts back on the baking sheet and top each with one of the burgers. Divide the onion mixture between the four burgers and top each one with two slices of cheese. Place the burgers back under the broiler to melt the cheese.
Serve the burgers with a knife and fork, and your favorite salad on the side.