Preheat oven to 425°F. Line baking sheet with foil. Place one potato flat on cutting board. Set chopsticks parallel on sides as slice guides.
Slice potato ⅛-inch thick crosswise using sharp knife, stopping at chopsticks for fan hinge. Repeat for all.
Pat potatoes dry with paper towels. Arrange fans on baking sheet standing up. Brush tops and sides with 4 tbsp melted butter using pastry brush till glistening.
Bake middle rack 30 minutes till skins crisp light gold and slices start fanning. Mix garlic, thyme, salt, pepper, remaining butter in bowl with spoon for baste.
Pull sheet. Brush garlic butter between every slice using pastry brush tip, work gentle. Sprinkle parmesan if using with measuring spoon. Bake 35-40 minutes more, basting every 15 minutes. Fans spread wide, edges deep golden.
Rest 5 minutes on sheet. Transfer to platter with spatula. Garnish extra thyme. Serve hot fanned tall.