Preheat the oven to 425˚F, and butter an 8" X 8" baking pan with 1/2 tablespoon of the unsalted butter. Lightly smear it with 1 teaspoon of the garlic paste.
Combine the milk, cream, broth, rosemary, salt, pepper, the remaining 1 1/2 tablespoons butter, and remaining 1 teaspoon garlic paste in a small pot over medium heat. Gently simmer for 5 minutes and allow to cool on the stove.
Slice the unpeeled potatoes on a mandolin the thickness of a quarter, about 1/8-inch. I used the thinnest setting on my hand-held mandolin, and sliced them all up in about 5 minutes. Layer the pan with a single layer of overlapping potato disks. Sprinkle with 1/3 of the grated cheese, and if desired, and little additional salt and pepper. Layer two additional layers of overlapping potato slices on top. Lightly sprinkle with salt and pepper if desired, and cover with half of the remaining cheese. Layer two final layers of potato slices, and top with the cheese.
Remove the rosemary sprigs from the liquid - don't worry if a few rosemary leaves detach into the liquid. Pour the warm liquid over the potatoes.
Place on a baking sheet, and slide into the oven. Bake for 30 minutes. Check it after 20 minutes to see if it's browning too quickly. If so, cover with foil, and continue to bake until the potatoes are tender.
Optional: Melt the optional butter in a small saucepan over medium-low heat, add the breadcrumbs, and toss to coat. Sauté on the stove until lightly browned. They'll brown more quickly the less you move them around. Sprinkle over the top of the baked potato gratin just before serving.
I like to use Yukon gold potatoes in a gratin since they have a great tendency to be soft and creamy than russets, but truthfully, you can use either type of potato. And search out ones grown in Idaho. The texture and flavor of Idaho-grown potatoes surpass any others.
This dish will keep in the refrigerator for a few days after cooking, and makes great leftovers. Likewise, it's a perfect dish to assemble ahead and bake off before the meal (just add an additional 10 minutes at least to the baking time), or bake it ahead and just warm back up in a 350˚F oven.