Prep: Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish with olive oil.
Slice: Using a mandoline or sharp knife, slice the sweet potato and butternut squash into uniform rounds.
Infuse Sauce: In a small saucepan over medium-low heat, add olive oil and garlic. Sauté for 1 minute until fragrant. Add milk, thyme, salt, pepper, and nutmeg. Simmer for 2 minutes (do not boil). Remove from heat.
Layer: Arrange the vegetable slices in the greased dish, overlapping them in an alternating pattern. Pour the warm milk mixture evenly over the top.
Bake Covered: Cover tightly with foil and bake for 40 minutes until vegetables are fork-tender.
Bake Uncovered: Remove foil, sprinkle mozzarella and parmesan cheeses on top, and bake for another 15-20 minutes until golden brown and bubbling.
Rest & Serve: Let rest for 10-15 minutes before serving to set the sauce.