Preheat your oven to 375°F (190°C). Butter a 9x13-inch baking dish generously and set aside.

Peel the potatoes and slice them 1/8-inch thin using a mandoline or a very sharp knife. Place slices in a large bowl of cold water as you work to prevent browning. When ready to assemble, drain and pat thoroughly dry with paper towels — removing excess moisture is key to a creamy (not watery) gratin.

In a large saucepan over medium heat, melt the butter. Add the minced garlic and cook, stirring, for 1–2 minutes until fragrant and just beginning to turn golden. Do not let it brown.

Whisk in the flour and cook for 1–2 minutes, stirring constantly, until the mixture smells slightly nutty and no raw flour taste remains.

Slowly pour in the warmed heavy cream and milk, whisking vigorously as you go to prevent lumps. Continue whisking over medium heat until the sauce thickens enough to coat the back of a spoon, about 4–5 minutes.

Reduce heat to low. Remove from heat briefly and stir in 2 cups of the shredded cheddar, a small handful at a time, whisking until fully melted and smooth before adding more. Stir in the Dijon mustard (if using), salt, pepper, nutmeg, and cayenne. Taste and adjust seasoning. The sauce should be boldly seasoned.

Spread a thin layer of the cream sauce across the bottom of the prepared baking dish. Arrange a single, slightly overlapping layer of potato slices across the bottom. Season lightly with salt and pepper. Spoon over enough sauce to just coat the layer. Repeat layering — potatoes, seasoning, sauce — until all potatoes and sauce are used, finishing with a generous pour of sauce over the very top.

Scatter the remaining 1/2 cup of sharp cheddar and the Parmesan over the top in an even layer.

Cover the baking dish tightly with aluminum foil. Bake covered at 375°F for 45 minutes — the potatoes should be just beginning to soften.

Remove the foil and continue baking uncovered for 20–25 minutes, until the top is deeply golden and bubbling vigorously around the edges, and a knife inserted in the center meets zero resistance.

If you want an even crispier, more browned top, switch the oven to broil for the final 2–3 minutes — watch it closely. Remove from the oven and let the gratin rest for 10–15 minutes before serving. This resting time allows the sauce to set up slightly for cleaner, more beautiful slices. Garnish with fresh chives and serve hot.