Line a rimmed baking sheet pan with plastic wrap.

Combine milk and salt in a saucepan over medium-high heat; bring to almost a simmer. Gradually whisk in semolina until mixture thickens, about 20 seconds. Reduce the heat to medium-low and continue stirring with a wooden spoon until very thick, 7 to 10 minutes.

Remove from the heat. Add 1 cup Parmigiano-Reggiano cheese, egg yolks, and cold butter; stir quickly to prevent yolks from cooking.

Transfer mixture to the prepared pan and spread out evenly. Cover with another piece of plastic wrap. Refrigerate until firm enough to cut, about 30 minutes.

Preheat the oven to 425°F (220°C). Grease a round, shallow baking dish.

Cut out circles of semolina dough with a 2 3/4-inch diameter cookie cutter. Use damp hands to gather and flatten any dough scraps. Press scraps between plastic wrap and cut as many more circles as possible.

Arrange gnocchi in a circular, overlapping pattern in the prepared baking dish. Drizzle melted butter over top and spread evenly with a brush. Sprinkle with cayenne pepper and cover with remaining Parmigiano-Reggiano cheese.