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 Tbsp. butter in a large skillet.
Grill sandwiches over medium heat until golden brown on one side.
Add remaining 1 Tbsp. butter to skillet, flip, and grill until golden brown on both sides.
*While sandwich is grilling, prepare sauce by melting 1 Tbsp. 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 2 extra large Phil's Fresh Eggs over easy.
Top each sandwich with 1 egg; sprinkle with fresh chives.