Place 1 ounce each of turkey, ham, and cheese on each of 6 bread slices; top with remaining 6 bread slices.
Combine eggs and half-and-half; stir well with a wire whisk. Carefully dip sandwiches into egg mixture, allowing excess to drip off.
Melt 1 tablespoon butter in a large nonstick skillet or griddle over medium-low heat. Place 3 sandwiches in skillet; cook 6 to 8 minutes on each side or until bread is golden and cheese melts. Remove from skillet, and sprinkle sandwiches evenly with half of powdered sugar. Repeat procedure with remaining 1 tablespoon butter, 3 sandwiches, and remaining powdered sugar.
Top sandwiches evenly with sour cream and strawberry preserves. Serve immediately.