1. To prepare grinder, place feed shaft, blade, and 1/4-inch die plate in freezer for 30 minutes or until well chilled. Assemble grinder just before grinding.
2. Arrange lamb pieces in a single layer on jelly-roll pan, leaving space between each piece. Freeze 15 minutes or until meat is firm but not frozen. Combine lamb and 1 tablespoon oil in large bowl; toss to combine. Pass lamb through meat grinder completely. Immediately pass meat through grinder a second time. Cover and chill.
3. Heat the remaining 1 tablespoon oil in a medium nonstick skillet over medium heat. Add onion and shallots; cook 15 minutes or until onions are golden, stirring frequently. Stir in 1 1/2 tablespoons garlic, cumin, and next 6 ingredients (through nutmeg); cook 1 minute. Remove from heat; cool to room temperature.
4. Combine lamb mixture, onion mixture, 1 tablespoon mint, and 1/4 teaspoon salt. Divide mixture into 4 equal portions, gently shaping each into a 1/2-inch-thick patty. Press a nickel-sized indentation in the center of each patty. Cover and chill until ready to grill.
5. Preheat broiler.
6. Cut bell peppers in half lengthwise; discard seeds and membranes. Place pepper halves, skin sides up, on a foil-lined baking sheet; flatten with hand. Broil 9 minutes or until blackened. Place in a zip-top plastic bag; seal. Let stand 10 minutes. Peel and cut each pepper portion in half.
7. Preheat grill to medium-high heat.
8. Combine remaining 1 1/2 teaspoons garlic, remaining 1 tablespoon mint, yogurt, juice, 1/4 teaspoon salt, and 1/8 teaspoon black pepper in a medium bowl. Set aside.
9. Sprinkle patties evenly with remaining 1/4 teaspoon salt and remaining 1/8 teaspoon black pepper. Place patties on a grill rack coated with cooking spray; grill 4 minutes or until grill marks appear. Carefully turn patties; grill 3 minutes or until desired degree of doneness. Place 1 patty on bottom half of each bun; top each serving with 2 tablespoons yogurt mixture, 1/2 cup radicchio, 2 bell pepper strips, and top half of bun.