How to Make It
Preheat oven to 475°F. Toss potatoes with 2 tablespoons oil, ￼¾ teaspoon salt, ½￼ teaspoon paprika, and several grinds of pepper on a rimmed baking sheet. Roast, tossing halfway through, until golden and tender, 20 to 25 minutes.
Heat remaining 1 tablespoon oil in a small nonstick skillet over medium-low. Add garlic, remaining ￼¾ teaspoon paprika, and ￼¼ teaspoon salt; cook until fragrant, about 30 seconds. Add tomato paste and stir to combine. Whisk in ￼½ cup water and simmer until reduced to ¼￼ cup, 8 to 10 minutes; transfer to a bowl. Let cool completely and whisk in mayonnaise.
Beat 2 eggs in a small bowl. Wipe out skillet and heat over medium-low. Add ￼¼ of the prosciutto and cook, turning once, until crisp, about 2 minutes. Add beaten eggs, tilting skillet to coat bottom. Top with 1 teaspoon chives and ¼￼ cup cheese. Cook until omelet is just set, 2 to 3 minutes. Fill with ¼￼ of the potatoes and transfer to a plate. Repeat with remaining prosciutto, eggs, chives, cheese, and potatoes to make 4 omelets.
Serve omelets topped with aioli and chives.