Preheat oven to 400°F. Butter a 12 cup muffin tin. Peel and finely dice the garlic and shallot. Place the diced garlic and shallot in a large bowl along with the half and half, 1 cup of grated Parmesan cheese, salt, pepper and thyme. Stir to combine and set aside.
Peel the potatoes and slice into 1/8? thick rounds with a knife or mandolin. Stack about 7 to 8 potato slices into each muffin tin. The number of slices will vary depending on the thickness of the cut. Stack them tightly but not any higher than the muffin tin itself.
Next, spoon the half and half mixture into each muffin tin over the potato slices, being careful not to overflow. Loosely cover the muffin pan with foil and place in the oven. Bake for 35 minutes. Remove the foil and carefully top each muffin tin with the remaining 1/2 cup Parmesan cheese. Place back in the oven without foil and bake for an additional 10 to 15 minutes until the cheese is golden brown, the potatoes are tender, and the liquid has absorbed.
Allow to cool in the pan for an additional 5 minutes, then carefully remove and serve. Add additional thyme sprigs as garnish if desired.