In a small bowl, mix mayo and dijon mustard until smooth.

Divide evenly between two slices of bread, then top each with 3 slices deli ham, 2 slices Swiss cheese, and remaining piece of bread.

Melt 1 Tablespoon butter in a large skillet.

Grill sandwiches over medium heat until golden brown on one side.

Add remaining 1 Tablespoon butter to skillet, flip, and grill until golden brown on both sides.

*While sandwich is grilling, prepare sauce by melting 1 Tablespoon butter in a small saucepan.

Stir in flour, then add milk and cook over medium heat, stirring until thickened and bubbly.

Stir in Swiss cheese until melted.

Line a rimmed baking sheet with foil.

Place grilled sandwiches on baking sheet, then pour sauce over sandwiches evenly.

Broil on low heat 2-4 minutes or until sauce is golden brown and bubbly.

*While sandwich is broiling, cook eggs over easy.

Top each sandwich with 1 egg; sprinkle with fresh chives.