Peel the potatoes and cut them into irregular cubes of about 2-3 cm. Pat them dry with a kitchen towel.
Heat the vegetable oil in a deep fryer or a large, heavy-bottomed pot to around 130°C (265°F). Add the potatoes and fry them for 10-12 minutes, or until they are cooked through but not yet crispy. Remove them and let them cool on a rack.
In a separate saucepan, heat the olive oil over medium heat. Sauté the chopped onion and garlic until soft. Stir in the sweet paprika, cayenne pepper (or chili flakes), and flour, and cook for a minute to form a paste (roux).
Slowly whisk in the tomato puree and vegetable stock, making sure there are no lumps. Bring the sauce to a simmer and let it thicken for 5-7 minutes. Season with salt and pepper to taste. Set aside.
Increase the oil temperature in your fryer to 180°C (350°F). Fry the par-cooked potatoes a second time for 3-5 minutes, or until they are golden brown and perfectly crispy. Remove them and drain on a paper towel. Season with salt immediately.
Serve the hot, crispy potatoes in a bowl. Pour a generous amount of warm brava sauce over them. Some people like to also add a drizzle of alioli sauce. Serve immediately.