Put the mayonnaise in a bowl and whisk in the lemon juice, garlic, olive oil, and salt to taste.  Let stand for at least 2 hours for the flavors to develop.

Place the potatoes in a large saucepan and cover with water.  Bring the pot to a boil over medium-high heat.  Reduce the heat to medium and cook until the potatoes are tender when pierced with a skewer, about 20 minutes.  Do not overcook.

Drain and let cool.  When the potatoes are cool enough to handle, cut each potato into quarters.

Preheat the oven to 475ºF.

Place the potatoes, 3 tablespoons of the olive oil and some salt in a bowl and toss to coat evenly.  Be careful not to break the potatoes apart, adding more oil if needed.  Spread the potatoes in an even layer on a large rimmed baking sheet.  Bake in the preheated oven, turning once, until crisp and browned, about 40 minutes.

While the potatoes are cooking, make the tomato sauce.

Heat the olive oil in a large heavy saucepan over medium heat.  Add the onion and garlic and cook until soft but not browned, 3 to 4 minutes.

Add the paprika, cumin and pepper flakes and stir for a few minutes.  Add the tomatoes with their juice plus 1/2 cup of water.  Break up the tomatoes (I use a potato masher – this makes it easy!)

Bring the mixture to a simmer and cook until the tomatoes have broken down, 15 to 20 minutes.  If the sauce gets too thick, add more water.  Add the vinegar and sugar.  Season to taste with salt and Tabasco sauce and remove from heat.

Let the sauce cool for a few minutes then using a blender or a food processor, puree the mixture.  Taste it again and add more salt or Tabasco if needed.  Serve the sauce at room temperature.

(You can store it in a jar, with a film of olive oil on top, in the refrigerator for up to 1 week.)

Arrange the potatoes on a serving platter and drizzle with the spicy tomato sauce and aioli.