Preheat oven to 400 degrees F.

In a Dutch oven or heavy pot with lid, place the three whole potatoes, the garlic head halves, fresh thyme and sea salt. Pour in enough water to come half way up the sides of the potatoes.

Bring to a boil, cover and place in the oven for one hour.

Remove potatoes to cool and discard the liquid and solids from the pot.

Once the potatoes are completely cool, preheat the oven to 450 degrees F.

Place a large skillet or sauté pan with the oil.

Bring heat to high and add in the whole rosemary. Leave rosemary in the oil for the entire frying time.

Slice the cooled potatoes into thick one inch slices and cook in two batches by frying in the hot oil, about four minutes per side. Remove to paper towels after frying. Also remove rosemary and discard stems but keep the fried rosemary pieces.

Lay out the cooked sliced potatoes into about nine stacks, cover with the cheddar, cooked rosemary pieces, scallions and Parmesan and bake in an oven-safe baking dish or pan, uncovered for ten minutes to melt the cheese.

Serve immediately.