Instructions

1. Arrange a rack in the middle of the oven and heat the oven to 400°F.
2. Prepare the following, placing each in the same large bowl as you complete it: Use a mandoline or sharp knife to thinly slice 3 medium russet potatoes and 2 medium sweet potatoes into 1/4-inch-thick rounds. Grate 4 ounces Gruyère cheese on the large holes of a box grater (about 1 cup). Grate 1 3/4 ounces Parmesan cheese on the small holes of a box grater or microplane (about 3/4 cup).
3. Grate 2 garlic cloves on the small holes of a box grater or on a Microplane. Pick the leaves from 4 large fresh sage sprigs and finely chop until you have 2 tablespoons. Pick the leaves from 7 to 8 fresh thyme sprigs until you have 1 teaspoon.
4. Melt 1 stick unsalted butter in a medium saucepan over medium-low heat. Add the garlic, sage, thyme, 2 1/2 teaspoons kosher salt, and 1/2 teaspoon black pepper. Stir to combine and cook until fragrant, about 1 minute. Pour over the potatoes and stir with a rubber spatula until the potatoes are thoroughly coated.
5. Sack the potato slices in the wells of a standard 12-well muffin tin, alternating the russet slices with the sweet potato slices and dispersing the cheese evenly throughout the layers, about 12 slices per stack. The stacks will be taller than the muffin tins; they will shrink as they cook.
6. Cover the stacks with aluminum foil and place the muffin tin on a baking sheet. Bake until the potatoes are slightly softened, about 20 minutes. Uncover and continue to bake until golden-brown and crispy, 20 to 25 minutes more. Let cool for 5 minutes. Run a butter knife around the edges of each stack to loosen, then remove the stacks from the tins. Sprinkle with flaky salt and serve immediately.