For the Pizza Onions:
2 tablespoons extra-virgin olive oil (EVOO)
2 large onions, quartered and thinly sliced
1 teaspoon dried oregano
About ¼ cup tomato paste
1 teaspoon Calabrian chili paste or crushed red pepper
½ cup red wine
1 cup chicken stock or water
For the Burgers:
1½ pounds ground beef sirloin
Kosher salt and black pepper
½ cup fresh flat-leaf parsley, finely chopped
6 tablespoons butter, at room temperature
3 cloves garlic, finely chopped
Drizzle of olive oil
4 slices each provolone and mozzarella, from the deli counter
4 Italian rolls or (4- to 5-inch pieces) soft Italian bread, split
Little gem lettuce leaves
For the pizza onions, in a large skillet, heat the oil, two turns of the pan, over medium. Add the onions; season with the salt and oregano. Cook, stirring often, until very soft, about 20 minutes. Add the tomato paste and chili paste and stir until coated, about 1 minute. Add the wine and stir until evaporated. Add the stock. Reduce heat to low. Let the pizza onions simmer while you make the burgers.
For the burgers, heat a large cast-iron skillet over medium-high.
In a large bowl, season the beef with kosher salt and black pepper and mix together. Form into 4 patties (thinner in the centers for even cooking).
In a small bowl, mix the parsley, butter, and garlic.
Drizzle the patties with oil. Cook, turning occasionally, 7 to 8 minutes. Top with the garlic butter, then the cheeses. Tent with foil to melt the cheeses.
Top the roll bottoms with lettuce, tomato, basil, and the cheeseburgers. Add the pizza onions and roll tops.