Preheat oven to 375°F (190°C).

Melt 1 tablespoon of butter in a pan over medium heat.

Add flour and whisk until a smooth paste forms.

Remove from heat and let cool for 2 minutes.

Gradually whisk in lukewarm milk until smooth.

Return the pan to medium heat, add mustard and nutmeg.

Simmer gently for 10 minutes, whisking frequently to prevent burning.

Remove from heat when the sauce thickens to tomato sauce consistency.

Add grated Gruyere cheese, reserving some for garnish.

Season with salt and pepper to taste.

Adjust consistency with more milk if needed; sieve if lumpy.

Remove crusts from bread slices and flatten with a rolling pin

Brush bread with melted butter.

Line muffin tin with buttered bread slices to create cups

Distribute ham evenly into bread cups.

Pour cheese sauce over ham.

Crack one small egg into each muffin cup.

Sprinkle reserved Gruyere cheese over the tops.

Bake for 20-25 minutes, or until eggs are cooked to your liking and golden brown.