Preheat a grill to high. Toss the chicken with 1 tablespoon olive oil, the paprika, 1/2 teaspoon salt and a few grinds of pepper in a medium bowl. Grill the chicken until marked and cooked through, about 7 minutes per side. Let cool slightly, then thinly slice.
Meanwhile, make the dressing Combine the garlic and the remaining 3 tablespoons olive oil in a small microwave-safe bowl; microwave until the garlic is softened, about 1 minute. Transfer the garlic and oil to a blender; add the 2 whole piquillo peppers and puree until smooth. Transfer to a bowl and whisk in the mayonnaise, lemon juice and 2 tablespoons cilantro; season with salt and pepper.
Divide the sliced chicken among the lettuce leaves; drizzle with the dressing. Top with the celery, almonds, the remaining sliced piquillo pepper and 1 tablespoon cilantro.