Heat 2 tablespoons olive oil in a skillet over medium heat. Remove the crust from 1 slice bread and tear the bread into 1-inch pieces. Add to the skillet along with the garlic and cook until golden brown, about 2 minutes. Add the almonds and cook until golden, about 3 minutes. Transfer to a mini food processor or blender along with the roasted peppers, vinegar, the remaining 2 tablespoons olive oil, 1/2 teaspoon paprika and 1/4 teaspoon salt. Pulse until slightly chunky.
Gently mix the ground meat, shallots, 1 teaspoon salt and the remaining 1/2 teaspoon paprika in a bowl with your hands until just combined. Form into four 1-inch-thick patties.
Heat a large nonstick skillet over medium-high heat. Brush with olive oil, then add the patties and cook until browned on the bottom, 4 to 5 minutes. Flip and top the patties with cheese; cook 4 to 5 more minutes. Spread the roasted pepper sauce on the remaining 8 slices bread. Serve the burgers on the bread with lettuce.