If grilling the burgers, heat your grill on high for at least 10 minutes.
To make the sauce, add all ingredients to a high-speed blender and blend until smooth.
To quick pickle the onions, add them to a small bowl and toss in the vinegar and salt. Set aside.
Form four (4-6 oz) ¼-½-inch thick patties (make them much wider than the buns). Season one side with salt and pepper to lightly coat.
Place the burgers, seasoned-side-down, on the grill or in a hot skillet with a drizzle of oil. Cook for about 2-3 minutes or until browned or you have nice grill marks. Sprinkle salt and pepper on the other side of the patty while it cooks. Flip, top with cheese, and cook, covered, for 2-3 more minutes or until cooked to your liking and the cheese is melted.
Place the buns on the grill or in the skillet to lightly toast them.
To assemble your burgers, place lettuce on the bottom bun, then add some pickled onions, then the patty, and top with a dollop of romesco sauce. Be sure to keep some extra sauce on your plate for dipping!