In a small saucepan over medium heat, melt 2 tablespoons of unsalted butter until bubbling. Whisk in flour, salt, white pepper, and nutmeg for about 1 minute until smooth. Gradually add milk, stirring until thickened. Remove from heat and mix in Gruyere cheese until melted.

Lay out two thick slices of country bread and spread Mornay sauce on each slice. Layer with ham and add Gruyere cheese on top. Spread Dijon mustard on remaining slices and place on top.

Heat a skillet over medium heat and place sandwiches inside. Toast for 2-3 minutes on each side until golden brown and cheese is melty, pressing down gently with a spatula.

Transfer to an oven-safe dish, spread remaining Mornay sauce on top, and sprinkle with Gruyere cheese. Broil on high for 2-3 minutes until cheese is bubbly and browned.

In another skillet, heat butter and crack in eggs, cooking until whites are set but yolks remain runny. Place a fried egg on top of each sandwich.