Step 1

Arrange a rack in the middle of the oven and heat to 400°F. Lightly grease a 1 1/2- to 2-quart baking dish with a drizzle of olive oil.

Step 2

Use a mandoline or chef's knife to slice the squash and potatoes into very, very thin slices, 1/8-inch thick or less. Place the sliced vegetables and olive oil in a large bowl and toss to combine.

Step 3

Place 1/3 of the squash and potato slices in and even layer in the bottom of the baking dish — no need to layer them squash-potato-squash; just spread them evenly — then season with salt and pepper. Sprinkle with 1/2 of the goat cheese in large chunks. Repeat with another 1/3 of the vegetables, seasoning again with salt and pepper and topping with the remaining goat cheese. Finish by layering on the remaining vegetables and seasoning with salt and pepper.

Step 4

Pour the milk evenly over the entire dish. Sprinkle with the Parmesan. Cover tightly with aluminum foil. Bake for 30 minutes. Uncover and bake until the top browns, about 15 minutes more. Sprinkle with the fresh basil or thyme, if using.