Preheat oven to 200°. Place a wire rack in a baking sheet and place in oven to preheat. In a large bowl, whisk eggs, milk, and salt until well combined.
Spread a thin layer of mustard on one side of each slice of bread. Layer half of bread slices with cheese, ham, turkey, and another slice of cheese. Close sandwiches with remaining bread.
In a large nonstick skillet over medium-low heat, heat 2 tablespoons butter until melted. Working 2 at a time, dip sandwiches into batter to coat, then place in skillet and cook, turning once, until golden and cheese is melted, 4 to 6 minutes per side. Transfer to rack in oven to keep warm. Repeat with remaining 2 tablespoons butter and sandwiches.
Divide sandwiches among plates. Dust with confectioners' sugar and serve with preserves alongside.