Cut potatoes into 1-in. chunks and put in a large pot. Add salt, sugar, and 2 qts. water; bring to a boil. Reduce heat to low and simmer potatoes until just tender, checking with a knife every 2 minutes (you don't want them to lose their shape or fall apart). Drain. Spread on a rack in a single layer and let cool to room temperature.
Pour enough canola oil into a 4- to 5-qt. pot so it is 3 in. deep. Insert a deep-fry thermometer and heat oil to 325° over medium-high heat. Meanwhile, set cooling racks in 2 rimmed baking pans.
Make sauce: In a food processor, pulse tomatoes to a coarse purée. Heat olive oil in a small saucepan over medium heat. Add onion and garlic. Cook, stirring often, until soft and pale gold, 3 minutes. Add spices and salt and cook until fragrant, 30 seconds. Reduce heat to low, add tomatoes and vinegar, cover, and simmer, stirring often, 5 minutes to let flavors mingle.
With a slotted spoon, lower one-third of potatoes into oil. Cook, stirring occasionally to keep potatoes from sticking, until they begin to turn golden, about 5 minutes. Transfer potatoes to baking pans, arranging them in a single layer. Allow oil to return to 325° and cook remaining potatoes. Let cool completely, about 45 minutes. Save 1/2 cup oil from pot.
Preheat oven to 425°. Line 2 baking sheets with parchment. Toss potatoes with reserved oil and spread out on baking sheets in a single layer. Bake, switching pans and stirring halfway through, until crisp and deep golden, 25 to 30 minutes. Sprinkle with salt.
Arrange half of potatoes in a shallow serving bowl and spoon on about half of sauce. Layer remaining half of potatoes on top and add remaining sauce. Serve with Aatxe Allioli.
*Smoked sweet Spanish paprika is available at well-stocked grocery stores. Find both types at Sur La Table and spanishtable.com.
Make ahead: Brava Sauce, up to 3 days, chilled airtight. Potatoes through step 4, up to 1 day, chilled airtight.