Combine eggs, 1/4 cup butter, French dressing, Worcestershire sauce, mustard, and salt; stir until well blended. Chill.
Trim crust from bread; cut into 2-inch rounds. Lightly spread whipped butter on one slice of white bread and one slice of wheat bread; spread egg salad on each slice. Stack bread, placing wheat on top, and cover with a white bread round. Slice sandwich in half vertically. Repeat with remaining bread and egg mixture.