Spread bread slices with mayonnaise and sprinkle with white pepper. Layer 2 slices with crab, avocado, onion, and cheese. Set remaining bread slices on top and spread each with 1 tsp. butter.
Melt remaining 2 tsp. butter in a large nonstick frying pan over medium heat. Put sandwiches in pan, unbuttered side down, and cook, turning once, until golden brown and cheese is melted, about 6 minutes total.
Note: Nutritional analysis is per sandwich.