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.
'What’s your Poblano?' Carne Asada Burger