How to Make It
In a small bowl, combine the mayonnaise, garlic, parsley and vinegar. Season with salt and pepper.
In a skillet, heat the 1 tablespoon of oil. Add the ham; cook over moderately high heat, turning once, until crisp, 2 minutes. Drain on paper towels.
Light a grill. Mix the pork, paprika, 1 tablespoon salt and 1/2 teaspoon pepper. Shape into four 3/4-inch-thick patties. Grill over moderately high heat for 5 minutes per side, or until just cooked through. Top with the cheese, cover and cook until melted. Transfer to a plate. Brush the cut side of the rolls with oil and grill. Spread the rolls with the mayonnaise, add the burgers and top with the piquillos and ham.