Make the potatoes: Put the potatoes in a large pot; cover with water by 1 inch and season with salt. Bring to a boil, then simmer until tender, 12 to 15 minutes. Drain the potatoes on a kitchen towel–lined baking sheet; let cool.
Working in two batches, fry the potatoes in 365˚ vegetable oil until crisp, 3 to 4 minutes. Drain on paper towels.
Meanwhile, make the aïoli: Soak the saffron in 3 tablespoons hot water, 10 minutes. Puree in a blender with the egg yolks, garlic, lemon juice, mustard and a pinch of salt.With the blender on high speed, slowly add the vegetable and olive oil and blend until thick and smooth, adding a few drops of water if needed; season with salt.
Sprinkle the potatoes with salt and smoked paprika and drizzle with the saffron aïoli.