Preheat the oven to 350 degrees F.
Heat 2 tablespoons of olive oil in a 10-inch ovenproof nonstick skillet over medium-high heat. Add the garlic, onion, paprika, cayenne and 1/2 teaspoon salt and cook, stirring, until the onions are soft and just starting to turn brown, about 6 minutes. Remove from the heat and stir in the vinegar until absorbed. Transfer to a bowl and let cool.
Whisk together the eggs, 1/2 cup of water and 1 teaspoon of salt in a large bowl. Add the potato chips and cooled onions and stir to combine. Let sit for a few minutes.
Heat the remaining 1 tablespoon oil in the skillet over medium-high heat, then add the potato and egg mixture and cook, until the bottom and the sides set and are a light golden brown, about 3 minutes (use a rubber spatula to help pull and peek around the sides). Transfer to the oven and bake until set, 15 to 20 minutes. Let cool slightly in the pan, then slide out onto a cutting board and cut into 12 wedges. Serve at room temperature.
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