Preheat the oven to 400°F. Lightly grease a large cast iron skillet, braiser or large casserole pan with butter and set aside.
In a small saucepan over low-medium heat, heat together the milk, butter, and garlic. Let it come to a low simmering boil, then allow to heat together for 1 minute. Turn off the heat and set aside.
Arrange the sliced sweet potatoes and Yukon gold potatoes so that they slightly overlap each other in circular fashion at the bottom of the prepared greased pan to create one layer that covers the bottom of the pan.
Sprinkle a little of the the thyme, paprika, salt, and pepper. Then sprinkle 1/3 of each of the cheeses over the layer. Then pour 1/3 of the milk mixture over the layer.
Repeat the layers in the same order - potatoes, spices, cheese, then milk mixture - two more times. You'll have 3 layers total and should have used all or most of the ingredients.
Cover, then bake at 400°F for 40 minutes. Then uncover and bake for an additional 15 minutes, or until the potatoes are fork tender.
Notes
Makes approximately 6-8 side servings.Make it dairy-free: use your favorite DF cheese shreds, vegan butter, and DF milk or cream. Storage: store leftovers in an airtight container in the refrigerator for up to 5 days. To reheat: reheat in a 325°F oven for about 10-15 minutes, until the center is heated through.