1. Preheat oven to 375°.
2. Remove white papery skin from garlic head (do not peel or separate cloves). Drizzle 1 teaspoon oil over garlic; wrap in foil. Bake at 375° for 1 hour; cool 10 minutes. Separate cloves; squeeze into a small bowl to extract pulp. Discard skins. Mash garlic using the back of a spoon. Stir in yogurt, juice, 1/4 teaspoon pepper, and 1/8 teaspoon salt.
3. Place 3 shrimp and 2/3 cup beans in a food processor; pulse until blended but not quite pureed. Add remaining shrimp, remaining beans, bulgur, cilantro, remaining 1/4 teaspoon pepper, and remaining 1/8 teaspoon salt to food processor; pulse until coarsely chopped. Fill a 1/4-cup dry measuring cup with shrimp mixture. Invert onto a platter lined with parchment paper; gently pat into a 2 1/2-inch-wide patty. Repeat procedure with remaining shrimp mixture, forming 8 cakes. Refrigerate 20 minutes.
4. Preheat broiler to high.
5. Brush a jelly-roll pan with 1 tablespoon oil. Arrange chilled cakes on pan; brush tops of cakes with remaining 1 tablespoon oil. Broil 5 minutes or until browned. Carefully turn cakes over. Brush tops of cakes with oil from pan. Broil an additional 5 minutes or until browned. Serve cakes with sauce.