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Preheat oven to 400 degrees F. Lightly grease a 1 1/2 to 2-quart casserole dish with baking spray. Set aside.
With a mandoline or sharp chef's knife, slice the potatoes into very thin slices, 1/8-inch-thick or less. Place the potatoes in a large bowl, drizzle with olive oil, then sprinkle with the minced garlic, salt, and pepper. Toss to coat the slices as evenly as you can.
Spread 1/3 of the potato slices in the bottom of the prepared dish. Crumble half of the goat cheese over the top. Repeat with the next 1/3 of the potatoes, then the remaining goat cheese, then finish by layering on the final third of the potatoes. The potatoes may discard some liquid as they rest in the bowl. If this happens, simply leave the liquid in the bottom bowl and shake the potato slices gently in your hands to remove excess liquid before layering them in the dish.
Pour the milk evenly over the top of the dish, then sprinkle with the Parmesan cheese. Cover the dish with foil, bake for 30 minutes, then uncover and bake for 15 additional minutes, until the top has browned. Scatter the rosemary over the top. Serve hot.