Preheat oven to 375F with rack on middle position.
In a bowl, whisk together the melted butter, olive oil, salt, and black pepper. Brush bottom of a 9-inch baking square or oval dish with a bit of butter mixture; set the remaining mixture aside.
Slice potatoes thinly and evenly cross-wise. For fastest and most even results, use a mandolin to slice potatoes. A sharp knife will work, too.
Arrange potato slices upright in prepared dish. Wedge chopped onions and garlic throughout potatoes. Brush with butter mixture, reserving 1/4 cup for later.
Bake 1 hour uncovered. Brush on the reserved 1/4 cup butter mixture and continue baking another 30 minutes until top is lightly browned. If it starts browning too quickly, cover loosely with foil.
Sprinkle top with shredded parmesan cheese and green onions. Bake 5-10 minutes until cheese is golden and melted. Sprinkle top with a light dusting of kosher salt and black pepper. Serve immediately.
Notes
Keep slices upright. Pack the potato slices snugly in the dish to help them hold their shape and bake evenly.
Brush generously. Coat the potatoes well with the butter and oil mixture for the best flavor and golden edges.
Use freshly shredded cheese. It melts more smoothly and gives a better crust than pre-shredded packaged cheese.
Serve right away. These potatoes taste best fresh from the oven when the edges are crisp and the centers are tender.
Variations: Try adding bacon bits, different cheeses, or fresh herbs like thyme or rosemary for extra flavor.
Storage and Reheating: Store leftovers airtight in fridge for up to 2 days. Warm leftovers uncovered at 350F in the oven just long enough to bring back the crisp texture.
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