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Step 1
Preheat oven to 425 degrees. Line a 13 by 9-inch baking sheet with aluminum foil (the foil should also run up the inside edges of the pan so the butter doesn't seep under, make sure there are no holes as well).
Step 2
Place two wooden chopsticks, skewers or wooden spatulas along sides of one potato (this is just to help cut all the way through). Slice potato into thin slices about 1/6th-inch thick but do not cut through the bottom 1/3-inch of the potato so the slices all stay intact.
Step 3
Repeat with remaining potatoes.
Step 4
Transfer potatoes to foil lined baking sheet. Brush all over and between slices with olive oil (just brush quickly between slices it doesn't need to be perfect).
Step 5
Bake in preheated oven 50 to 55 minutes until fairly tender (potato slices should be falling open).
Step 6
Near the end of that time mix together melted butter, garlic and thyme.
Step 7
Once time is up remove potatoes from the oven and pour and brush the butter mixture over potatoes (basting between the slices as well. You can brush with butter that's now on bottom of baking sheet. Note that there will be some butter on bottom of sheet as they bake and this is fine).
Step 8
Season with salt and pepper (again getting some between the slices). Sprinkle parmesan over potatoes and between slices.
Step 9
Return to oven and continue to bake until centers of potatoes are fully tender, about 15 to 20 minutes longer.
Step 10
Brush with butter from the baking sheet, sprinkle with parsley, serve warm.