Put the potatoes in a 5-quart / 4.7 L pot, cover with cold water by 1 inch / 2.5 cm, and add 1 tbsp kosher salt. Bring to a boil, then simmer for 6 to 8 minutes, until the edges look slightly fuzzy and a knife slides in with some resistance at the center.

Drain gently and spread the potatoes in a single layer on a wire rack set over a 33 x 46 cm / 13 x 18 inch sheet pan. Let them steam-dry for 10 minutes, until the surface looks matte and dry.

While the potatoes dry, heat the olive oil in a 2-quart / 1.9 L saucepan over medium-low heat. Cook the garlic for 30 to 60 seconds until fragrant but not browned, then stir in the flour for 1 minute. Pull the pan off the heat, stir in both paprikas and the tomato paste, and return it to the heat. Pimentón scorches in seconds, so if it smells acrid or turns dark brown, start over.

Whisk in the stock a little at a time, then add the sherry vinegar and 1/2 tsp salt. Simmer gently for 8 minutes, until the sauce coats the back of a spoon. For the optional alioli, stir the mayonnaise, grated garlic, and lemon juice together in a small bowl.

Pour the sunflower oil into a 24 cm / 9.5-inch Dutch oven, about 1.5 inches / 4 cm deep, and heat to 350 F / 175 C. Fry half the potatoes (about 450 g) for 7 to 8 minutes, turning now and then, until deep golden and crisp. Lift them onto the wire rack with a spider skimmer and sprinkle with half the flaky salt.

Let the oil return to 350 F / 175 C, then fry the second batch the same way and salt it while hot.

Pile the potatoes on a platter or in a shallow dish, spoon the warm sauce over the top, and drizzle with the alioli. Serve within 10 minutes of frying.