Preparation

• In a food processor, pulse the red peppers, olives, garlic, parsley, and olive oil into a coarse paste, 15 to 20 one-second pulses. Set aside.
• Combine the goat cheese, mint, and lemon zest in a large bowl. Transfer about one-quarter of the mixture to a small bowl and set aside at room temperature. Refrigerate the goat cheese mixture left in the large bowl until firm, 10 to 15 minutes.
• Add the lamb, onion, egg, breadcrumbs, rosemary, 1/2 tsp. salt, and 1/2 tsp. pepper to the chilled goat cheese mixture and mix by hand until well combined there should be some small pieces of goat cheese visible. Shape into four 4-inch patties and transfer to a plate. Press 3 fingers down into the middle of each patty to create a shallow indentation, loosely cover with plastic wrap, and refrigerate until cold, at least 20 minutes and up to 3 hours.
• Heat a grill pan over medium-high heat or prepare a medium-high (400°F) gas or charcoal grill fire. Lightly oil the pan (or grill grates) and then cook the burgers, flipping once, until done to your liking, about 7 minutes total for medium (135°F to 140°F).
• Spread the remaining goat cheese mixture on one cut side of each roll and then assemble the burgers on the rolls with the lettuce and red pepper relish.