Whisk together eggs and next 2 ingredients in a shallow dish. Dip both sides of each bread slice into egg mixture.
Melt 1 Tbsp. butter in a large nonstick skillet over medium heat; add 3 bread slices, and cook 2 to 3 minutes on each side or until golden brown. Remove from skillet. Repeat with remaining 1 Tbsp. butter and 3 bread slices.
Place bread slices on a baking sheet. Top each bread slice evenly with ham and cheese.
Broil 5 1/2 inches from heat 3 to 5 minutes or until cheese is melted. Serve immediately with dill pickle spears.