Preheat oven to 400 degrees. Lightly grease a 1 1/2 to 2 quart casserole dish with a drizzle of olive oil.
Use a mandolin or chef's knife to slice the potatoes into very, very thin slices, 1/8 inch or less. Toss the sliced vegetables with the 3 T. of olive oil in a large bowl.
Place 1/3 of the squash and potato slices in the bottom of the dish--no need to layer them squash-potato-squash---just spread evenly. Season with salt and pepper. Top with 1/2 of the goat cheese, scattered evenly in large chunks. Repeat with another 1/3 of the vegetables, seasoning again with salt and pepper and topping with the other 1/2 of the goat cheese. Finish by layering on the final 1/3 of the vegetables and seasoning with salt and pepper.
Pour the milk over the entire dish and then top with parmesan cheese. Bake covered for 30 minutes and then uncover and bake 15 minutes or until the top browns. Scatter on the fresh basil or thyme.