1. Combine first 3 ingredients in a saucepan; bring to a boil. Reduce heat; simmer 30 minutes or until mushy. Discard thyme. Drain, pressing to remove excess water. Cool slightly.
2. Combine black pepper, paprika, and red pepper. Combine yogurt, mayonnaise, pickles, and mustard.
3. Place crab in a medium bowl; mash slightly. Add quinoa, 1/2 teaspoon spice mixture, half of yogurt mixture, bell pepper, and next 4 ingredients (through egg white); stir gently. Divide mixture into 8 equal portions; gently pat into a 3-inch-wide patty. Place on a parchment-lined plate. 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 with remaining 1 tablespoon oil. Sprinkle with half of remaining pepper mixture. Broil 5 minutes or until browned. Turn cakes over. Brush with oil from pan; sprinkle with remaining spice mixture. Broil 5 minutes or until browned. Serve cakes with remaining sauce.