Prep the potatoes
Bring a large pot of generously salted water (it should taste like the ocean) to a boil over high heat. While the 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, transfer it to a cup. Add another egg and a teaspoon of mustard to the empty blender and blend, then slowly drizzle the broken aioli back in while the 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; 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 the second batch. Transfer the potatoes to a serving platter and dollop with the bravas sauce and aioli.
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