Urdaburger

Stir together cola, red wine, piquillo peppers, shallot, garlic, and piment d’Espelette in a small saucepan. Bring to a boil over medium, and cook, stirring occasionally, until thickened to a syrupy consistency and reduced to about 1/4 cup, 18 to 20 minutes. Set aside to let cool completely, about 45 minutes. Process cola mixture, mayonnaise, sherry vinegar, brown sugar, and salt in a blender until smooth, about 15 seconds. Transfer to a small bowl; cover and chill until ready to use, up to 3 days.

Stir together all ingredients in a medium bowl until well combined. Cover and chill.

Arrange jamón serrano in a single layer on a microwavable plate lined with paper towels. Cover with a paper towel. Microwave on HIGH until slightly crisp, 1 to 2 minutes. (Ham will crisp as it cools.)

Preheat grill to high (450°F to 500°F). Let hamburger patties stand at room temperature while grill is preheating. Shape pimentón cheese into 4 (3-inch) disks (about 1/4 cup each). Press a dimple into the center of each hamburger patty to prevent shrinking when cooking, and season evenly with salt and pepper. Place patties on oiled grates; grill, uncovered, until lightly charred, 3 to 4 minutes. Flip patties, and top each with 1 pimentón cheese disk and 1 crispy jamón slice. Cook until burgers reach desired degree of doneness, 3 to 4 minutes for medium-rare, covering grill during last minute of cook time to melt cheese. 

Spread about 1 tablespoon aioli on each bun half. Layer bottom buns with lettuce and burger patties; top with pickles, and cover with bun tops. Serve immediately.