In a medium saucepan over medium heat, melt 3 tbs unsalted butter. Add 3 tbs all-purpose flour and whisk constantly for 1–2 minutes until the mixture smells slightly nutty and is a pale blond color (do not let it brown).

Add milk and thicken

Slowly whisk in 2 cups warm whole milk, a little at a time, to prevent lumps. Bring the sauce to a gentle simmer, whisking continuously, until it thickens and coats the back of a spoon (about 4–6 minutes).

Season and finish

Whisk in 1 tsp Dijon mustard, 1 tsp sea salt, 1/2 tsp freshly ground black pepper, and 1 tsp freshly grated nutmeg. Remove from heat and stir in 1/2 cup finely grated Parmesan until melted and smooth. Fold in 1 tbs finely grated truffle or truffle paste and taste; adjust salt and pepper if needed. Keep warm on very low heat, covered, stirring occasionally so it doesn't skin over.

Sandwich Assembly and Cooking

Prepare bread and cheese

Preheat oven to 425°F (220°C). Lightly butter one side of each of the 8 bread slices with softened butter (about 2 tbs used here). On the unbuttered side of four slices, spread a thin layer (about 2–3 tbs each) of the warm truffled béchamel. Sprinkle about 1/2 cup grated Gruyère evenly over the béchamel on each slice. Place 2 slices of ham on top of the cheese, then add another 1/4 cup Gruyère over the ham. If using fresh shaved truffle, shave a few thin slices over the ham at this stage.

Close and prepare for finish

Top each assembled slice with the remaining bread slice, buttered side up. Spread a thin layer (about 1–2 tbs) of béchamel on the top of each sandwich, then sprinkle the remaining grated Gruyère evenly across the tops (about 2–3 tbs each). This will create a gratinée crust when baked.

Bake to gratinée

Place sandwiches on a baking sheet lined with parchment or a lightly buttered ovenproof skillet. Bake in the preheated oven for 8–10 minutes until the cheese begins to melt, then switch the oven to broil (high) and broil 2–4 minutes, watching carefully, until the top is deeply golden and bubbling. Remove from oven and let rest 1–2 minutes.

Pan-toast alternative (optional)

If you prefer a crisper bottom, finish the broiled sandwiches in a hot skillet: heat 1 tbs butter in a skillet over medium-high heat and brown each sandwich bottom for 30–60 seconds before broiling, or pan-fry assembled sandwiches 2–3 minutes per side on medium until golden then finish under the broiler for 1–2 minutes to melt the top.

Eggs and Finishing

Fry eggs

While sandwiches are in the oven, heat 1 tbs unsalted butter in a nonstick skillet over medium heat. Crack 4 large eggs into the pan and cook sunny-side up until whites are set but yolks remain runny, about 2–3 minutes. Season each egg with a pinch of salt and 1/8 tsp black pepper. For slightly set yolks, cover the skillet for the final 30–45 seconds.

Assemble and garnish

Place one hot Croque sandwich on each plate. Spoon any remaining warm truffled béchamel over the top if desired. Carefully place a fried egg on top of each sandwich. Drizzle each egg lightly with 1/4 tsp truffle oil (optional). Garnish with microgreens or chopped parsley and additional shaved truffle if available.