Parboil the potatoes approximately 5 to 6 minutes in salted water. Drain and cool to room temperature. Meanwhile, saute the diced onions in the oil plus 1 tablespoon of the butter until richly browned and caramel colored. This step ensures that all of the onion's sweetness is developed. Cool the onions, then mix with the potatoes.
Preheat the oven to 350 degrees. Melt half the remaining butter in a 10-inch nonstick saute pan over medium heat, and add half the diced potato/onion mixture. Take care to pack the mixture tightly in one neat layer to saute and brown. Using the back of a spoon, form the potato and onion mixture into a rounded cake. When the potatoes and onions begin to adhere and brown, flip them over using a large spatula or the back of a plate. Brown the second side, then transfer the hash-browned potatoes to a cookie sheet and reserve while you cook the second half of the potato/onion mixture. Add the second batch to the cookie sheet and bake 10 minutes. Remove from the oven, season with the salt and pepper, and serve.