Bring the milk, salt, and nutmeg to a gentle simmer in a heavy-bottomed saucepan.

Reduce heat to low and whisk in the semolina in a slow, steady stream to avoid lumps.

Switch to a wooden spoon and cook, stirring often, for 8 to 10 minutes until the mixture pulls cleanly from the sides of the pan and looks like thick porridge.

Remove from heat. Stir in the egg yolks, half the butter, and half the Parmigiano until smooth.

Spread the mixture about 1 cm thick onto a buttered sheet pan or marble slab using a wet spatula.

Let it cool at room temperature, then chill for at least 30 minutes until firm enough to cut cleanly.

Cut into rounds with a 6 to 7 cm cutter dipped in warm water between cuts.

Heat the oven to 220 C / 425 F.

Arrange the discs overlapping like roof tiles in a buttered baking dish.

Dot the top with the remaining butter and scatter the remaining Parmigiano over everything.

Bake for 15 to 20 minutes until the tops are golden with browned spots and the cheese has crisped.

Let rest 5 minutes before serving.