Preheat oven to 350°F (175°C) and grease a 9-inch tart or pie pan with butter.
Roll out and fit the pie crust into the pan; trim and flute edges. Chill 10 minutes if needed.
Optional: brush the crust lightly with Dijon mustard for extra flavor.
Scatter diced ham evenly over the bottom of the crust.
Sprinkle grated cheese over the ham, leaving room near the edges.
Whisk together eggs, milk or half-and-half, salt, pepper, and nutmeg until smooth.
Pour the egg mixture slowly into the crust, allowing it to settle around the fillings.
Place the quiche on a baking sheet to catch any drips and bake in the preheated oven.
Bake 35–45 minutes until the center is just set and the top is lightly golden; edges should be firm.
Cool 10–15 minutes before slicing to let the custard finish setting for clean slices.