In a small sauce pan, melt the butter. Whisk in the flour and cook, whisking constantly, 2 minutes. Do not let the flour brown. Whisk in the milk and add the rosemary. Cook, stirring occasionally, until thick, about 10 minutes. Remove the pan from the heat and discard the rosemary. Whisk in the mustard, nutmeg, salt and Espelette pepper. Keep warm over very low heat.

Butter one side of each bread slice. Spread the other side of four of the slices with about 1 tablespoon Bechamel and top each with one-quarter of the ham. Sprinkle one-eighth of the Gruyere on top of each. Cover each sandwich with a remaining bread slice with the buttered side facing out.

Warm a large skillet over medium-high heat. Grill each sandwich until both sides are toasted and golden brown, about 2 minutes per side. Transfer the toasted sandwiches to a rimmed baking sheet and rub both sides of each with garlic. The toasted bread will gently grate the garlic.

Preheat the broiler to high. Spread 2 tablespoons béchamel on top of each sandwich. Sprinkle the remaining Gruyere evenly over the sandwiches. Broil until the cheese is bubbly and golden brown, 2 to 3 minutes.

In a large skillet over medium-high heat, warm the oil. Crack the eggs into the skillet and fry until the white of each is cooked through for sunny-side-up, about 3 minutes, or flip and cook the eggs on the other side for over-easy. Top each sandwich with a fried egg, season with salt and pepper, and serve at once.