Step 1

Place potatoes and parsnips in a bowl of cold water.

Step 2

Preheat oven to 375 degrees F (190 degrees C). Place garlic and melted butter into a 2-quart baking dish; brush garlic and butter all over the inside of the dish.

Step 3

Slice potatoes very thin using a mandoline slicer.

Step 4

Layer 1/3 potato slices into the bottom of the prepared baking dish. Season with salt and black pepper. Sprinkle a few thyme leaves onto potato slices and a light dusting of Parmigiano-Reggiano cheese. Spoon about 3 tablespoons creme fraiche over cheese.

Step 5

Thinly slice parsnips using a vegetable peeler; arrange 1/2 the parsnip slices in an even layer over the creme fraiche and season with salt and black pepper. Repeat potato and parsnip layering process: 1/3 potato slices, salt and black pepper, thyme, Parmigiano-Reggiano cheese, creme fraiche, and remaining 1/2 parsnip slices. Dust with salt. Arrange last 1/3 potato slices on top and season with salt.

Step 6

Pour chicken broth over potato-parsnip layers about 3 tablespoons at a time. Shake dish gently to eliminate air bubbles. Gently spread remaining 2 tablespoons creme fraiche over the potatoes. Sprinkle cayenne and remaining Parmigiano-Reggiano cheese over the top.

Step 7

Bake in the center of the preheated oven until gratin is browned and a knife pierces layers easily, 45 minutes to 1 hour.