Prepare the grill for indirect cooking over high heat (450° to 550°F).
Cut off and discard the top third of the head of garlic to expose the cloves. Place the garlic on a square of aluminum foil and drizzle with 2 teaspoons oil. Fold up the sides and seal to make a packet, leaving a little room for the expansion of steam.
Brush the cooking grates clean. Place the garlic packet over indirect high heat, and place the bell pepper, the tomato, and the bread over direct heat. Close the lid and cook until the garlic is soft, 30 to 45 minutes; the bell pepper is blackened and blistered all over, 10 to 15 minutes, turning occasionally; the tomato is softened, 8 to 10 minutes, turning once or twice; and the bread is toasted, about 1 minute, turning once. Remove from the grill as they are done. Place the bell pepper in a bowl and cover with plastic wrap to trap the steam. Let stand for about 10 minutes. Remove and discard the charred skin, stem, and seeds from the pepper. Remove any loose skin from the tomato, cut in half, and remove the core. Tear the bread into pieces. Open the garlic packet and let cool for about 10 minutes.
In a food processor or blender combine the almonds with the vinegar and process until the almonds are finely chopped. Squeeze out half of the garlic cloves (save the remaining garlic for another use) and add to the food processor along with the bell pepper, tomato, bread, crushed red pepper flakes, ¼ teaspoon salt, and ⅛ teaspoon pepper. Process until the sauce has the consistency of natural peanut butter. While the food processor is running, add ¼ cup oil in a slow and steady stream. Set aside.
Gently combine the lamb, paprika, cumin, 1 teaspoon salt, and ½ teaspoon pepper and form into four equal-sized balls. Make a deep indentation in the center of each ball of meat, and insert one-quarter (about 2 tablespoons) of the cheese inside. Close the meat over the cheese and gently form into patties of equal size, each about ¾ inch thick, being sure the cheese is completely covered. Lightly brush the patties on both sides with oil.
Change the method and temperature of the grill to direct cooking over medium heat (350° to 450°F).
Grill the patties over direct medium heat, with the lid closed, until cooked to medium doneness (160°F), 8 to 10 minutes, turning once (some cheese may escape). During the last minute of cooking time, toast the buns, cut side down, over direct heat.
Build the burgers on buns with lettuce, a patty, and ¼ cup of the sauce. Serve immediately. Reserve any remaining sauce for another use.