Preparation

• Set your oven to 375°F and position a rack in the center.
• In a medium skillet over medium heat, melt 3 tablespoons of butter until it's just bubbling. Add the gluten-free all-purpose flour and whisk continuously for about 30 seconds to create a roux. Gradually pour in the milk while whisking to prevent lumps, then cook until the sauce thickens enough to coat the back of a spoon (about 4 minutes). Remove from heat and stir in the Dijon mustard, salt, and nutmeg.
• Spread softened butter generously on one side of each bread slice. Place four slices buttered-side down on a baking sheet, then layer with ham and a generous portion of shredded Gruyère on each. Top with the remaining bread slices, buttered-side up.
• Slide your baking sheet into the oven and bake for about 6 minutes until golden-brown and the cheese is starting to melt.
• Remove sandwiches from the oven and spread the béchamel sauce evenly over the top of each one. Switch your oven to broil and return the sandwiches until the cheese bubbles and develops golden-brown spots, about 3-4 minutes.
• While your sandwiches broil, heat a bit of butter or oil in a skillet over medium heat. Crack your eggs into the pan and fry gently until the whites are set but the yolks remain runny (about 5 minutes).
• Carefully place one fried egg on top of each sandwich, then garnish with chopped fresh chives and black pepper. Serve immediately.