One 14-ounce can whole tomatoes with juices, chopped
1 garlic clove, halved
2 teaspoons fresh lemon juice
1 large egg yolk, at room temperature
1/2 cup plus 2 tablespoons extra-virgin olive oil
Extra-virgin olive oil, for frying
2 pounds baking potatoes, peeled and cut into 1-inch cubes
In a medium skillet, heat the olive oil. Add the crushed red pepper and cook over moderately high heat until fragrant, about 30 seconds. Add the tomatoes and their juices and boil until the sauce is reduced to 3/4 cup, about 5 minutes. Season with salt and transfer to a serving bowl.
Set a small bowl over a damp dishcloth to prevent it from sliding. In a mortar, pound the garlic to a paste. Transfer the paste to the bowl. Add the egg yolk and 1 teaspoon of the lemon juice and whisk to combine. Gradually add half of the olive oil in a thin stream, whisking until the aioli is very thick. Whisk in the remaining 1 teaspoon of lemon juice. Add the remaining olive oil in a thin stream, whisking until thick. Season the aioli with salt and transfer to serving bowl.
Set a rack over a large rimmed baking sheet. In a large, deep skillet, heat 1/4 inch of olive oil. Add the potatoes and fry over moderate heat, turning once, until just softened, about 7 minutes. Increase the heat to high and cook, turning once, until golden and crisp, about 5 minutes. With a slotted spoon, transfer the potatoes to the rack to drain. Season the potatoes with salt.
Arrange the potatoes in the center of a platter. Place the sauces on either side and serve right away.
The tomato sauce and aioli can be refrigerated overnight. Bring to room temperature before serving.