Step 1

In a large pot of boiling water, cook all of the potatoes with a large pinch of salt until fork tender, about 20 minutes. Drain and return the potatoes to the pot. Add the cayenne, nutmeg and 2 tablespoons of the stock. Mash the potatoes until creamy and almost smooth, adding more broth if needed. Season with salt, then spread the mashed potatoes evenly in a medium baking dish.

Step 2

Meanwhile, preheat the broiler with the rack 6 inches from the heat. In a large, nonstick skillet, heat the olive oil. Add the shallots and cook over moderate heat, stirring occasionally, until softened and light golden, 5 to 8 minutes. Scatter the shallots over the potatoes, then top with the cheese.

Step 3

Broil the gratin until the cheese is melted and the shallots are crisp, 2 to 3 minutes. Serve immediately or keep warm in a 300° oven.