Melt 4 tablespoons of unsalted butter in a 12-inch nonstick fry pan over medium heat.

Add the leeks and kosher salt, stirring to coat the leeks with butter.

Cover and cook, stirring occasionally, until the leeks are tender, about 20 minutes.

Add thyme, nutmeg, pepper, and heavy cream.

Simmer, uncovered, stirring occasionally, until the mixture has thickened, about 15 minutes.

Transfer the leek mixture to a bowl and let it cool.

Wipe out the pan and grease it with 1 tablespoon of butter.

Preheat the oven to 400°F (200°C).

In a separate bowl, combine the grated Gruyere and Parmigiano-Reggiano cheeses.

Layer one-third of the potato slices in the prepared fry pan.

Spread one-third of the cooled leek mixture on top of the potatoes.

Sprinkle with one-third of the cheese mixture, then sprinkle with 1 tablespoon of minced fresh chives.

Repeat the layering process two more times, reserving 1 tablespoon of chives.

Cover the pan tightly and transfer it to the preheated oven.

Bake for 45 minutes.

Remove the lid and continue baking until the potatoes are tender and the crust is golden brown, about 30 minutes longer.

Sprinkle the remaining 1 tablespoon of minced fresh chives on top.

Let the gratin stand for 15 minutes before serving.