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1
Heat the oven to 400 degrees F and arrange a rack in the middle.
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2
Pierce the potatoes all over with the tip of a paring knife and place directly onto the oven rack.
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3
Bake until barely tender when pricked with the point of a small knife, about 45 minutes to 1 hour.
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4
Let the potatoes cool to room temperature, then chill in the refrigerator for at least 2 hours, preferably overnight.
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5
Warm 1 tablespoon of the olive oil in a large, nonstick, ovenproof frying pan over medium heat.
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6
Fry the onion, stirring occasionally, until its lightly browned and tender, about 10 minutes.
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7
Stir in the salt, paprika, sugar, onion powder, and garlic powder.
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8
Scrape the onion into a large bowl.
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9
Peel the potatoes and shred them on the large holes of a box grater into the bowl with the onions.
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10
Toss the shredded potatoes with the onion mixture until well combined.
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11
Warm the frying pan over medium heat and add 2 tablespoons of the oil.
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12
When its hot, add the potato mixture.
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13
Using a spatula, quickly spread the mixture into an even, thick layer, then press down firmly on it to form a cake.
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14
Cook, pressing down occasionally with the spatula, until the bottom is crisp and golden brown, about 10 minutes.
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15
Set a large plate upside down over the top of the potato cake.
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16
Quickly (but carefully) invert the potatoes onto the plate.
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17
Add the remaining 1 tablespoon of oil to the pan and, using the spatula as a guide, slide the potatoes back into the pan so that the cooked side is on top.
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18
Cook, pressing down occasionally with the spatula, until the bottom is crisp and golden brown, about 10 minutes.
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19
Heat the broiler and arrange a rack about 6 inches from the heat.
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20
Carefully invert the potato cake onto a baking sheet.
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21
Sprinkle the top with the cheese and slide it under the broiler.
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22
Broil until the cheese is melted and lightly browned, about 2 minutes.
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23
Sprinkle with the green onions, cut into wedges, and serve.
