1 tablespoon sherry vinegar (red wine vinegar works also in a pinch)
½ teaspoon cayenne pepper
1 bay leaf
1 cup neutral-flavored oil, such as canola, vegetable, grapeseed
1 tablespoon dijon mustard
1 teaspoon sherry vinegar (lemon juice works in a pinch as well)
1 tablespoon minced garlic
½ teaspoon kosher salt, plus more to taste
3 liters canola oil, for frying
Prep the potatoes: Bring a large pot of generously salted water (it should taste like the ocean) to boil over high heat. While water is boiling, peel and dice the potatoes into ¾- inch cubes and cover with cold water in a large bowl to prevent them from turning brown. Once the water is boiling, lift the potatoes out of the cold water, add them to the boiling water, return to a boil and cook until the potatoes can be pierced with a skewer but are still al dente, 10 to 12 minutes. Drain, then arrange in a single layer on a paper towel-lined rimmed baking sheet to cool.
Make the bravas sauce: Warm the olive oil over medium heat in a small saucepan. Add the garlic and cook, stirring, until fragrant and lightly golden, 2 minutes. Add the tomatoes, salt, smoked paprika, vinegar, cayenne, and bay leaf, bring to a simmer and cook over medium-low heat until the sauce thickens, 30 minutes. Remove the bay leaf, season with more salt, and cover with foil to keep warm.
Make the aioli: Combine the egg, 2 tablespoons of the oil, mustard, vinegar, garlic, and salt in a blender or small bowl of a food processor and blend. With the motor running, drizzle in the rest of the oil in a very thin stream (this should take at least a minute) to create a thick, mayonnaise-like texture. Season with more salt to taste.Note: If aioli “breaks” during the emulsification process, no worries. Transfer it to a cup, then add another egg and a teaspoon of mustard to the empty blender and blend. Slowly drizzle the broken aioli back in while motor is running to emulsify.
Fry the potatoes and plate the dish: Heat the oil in a large (at least 5 quart) pot to 350°F. (Use a candy thermometer, or test the oil by dropping in a piece of sandwich bread into the oil; if it sizzles and begins to crisp and brown right away, it’s ready). Line a rimmed baking sheet with paper towels. Working in 2 batches, gently lower the potatoes into the oil and fry, stirring during the first minute so they don’t stick together, until golden brown and crisp, 6 to 7 minutes. Use a spider to lift the potatoes onto the towel-lined sheet and sprinkle with salt. Repeat with second batch of potatoes. Put the potatoes on a serving platter and dollop with the brava sauce and aïoli.