Braise the duck breasts for 5 minutes.
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.4 1/2-inch-wide.
grass-fed ground beef patties.(4-inch) potato hamburger buns, buttered and toasted.
Shredded iceberg lettuce and pickles, for serving.Make the kalimotxo aioli.