Heat oven to 350. Coat a 13 x 9 inch baking dish with butter; set aside.

Place half-in-half, garlic, salt, pepper and thyme in a large, wide pot or Dutch oven. Peel and slice potatoes into 1/8-inch thick slices (a mandolin works great in this recipe) and add them to the pot. Bring the mixture to a simmer over medium heat, gently folding the potatoes into the sauce occasionally, about 20 minutes total.

Using a slotted spoon, transfer half of the potatoes to prepared baking dish and arrange them in an even layer. Evenly sprinkle with half of the cheese and half of the cubed ham.

Transfer the remaining potatoes to the dish, arranging in an even layer. Pour the half-in-half mixture from the pot over the potatoes. Sprinkle with the remaining half of the cheese and cubed ham. Scatter the butter pieces on top.

Bake uncovered until the potatoes are very tender and the top of the casserole is golden brown and bubbling, about 1 hour. (Cover the dish with foil if the top starts to get too brown)

Remove to a wire rack and let sit 10 minutes before serving.