To make the sauce, combine the oil, garlic, and chilies in a small saucepan. Cook over medium heat, stirring, until fragrant, about 15 seconds. Add the tomato paste and sherry vinegar. Cook, stirring, until the vinegar has evaporated, about 1 minute. Add the tomatoes and sugar and reduce the heat to low. Cook, stirring occasionally, until thickened and the flavors have melded together, about 30 minutes. Stir in the pimentón and salt.
Transfer to a blender or food processor and puree until very smooth, scraping the bowl occasionally. Add water, 1 teaspoon at a time, until the mixture achieves a ketchup-like consistency that you can drizzle but is still quite thick. Keep at room temperature if serving immediately. The sauce can be refrigerated in an airtight container for up to 2 weeks.
To make the potatoes, preheat the oven to 475°F with a half-sheet pan on the center rack.
Toss the potatoes with the oil and 1 teaspoon salt in a large bowl. When the oven has fully heated, carefully spread the potatoes on the hot pan in a single layer, scraping all the oil in the bowl onto the pan.
Roast until the bottoms are deeply browned and release easily from the pan, about 20 minutes. Carefully flip the potatoes and roast until the other sides brown, 10 to 20 minutes. Sprinkle with ½ teaspoon salt and transfer to a serving platter.
Serve with the brava sauce and allioli, either in bowls for dipping or by squirting all over the top using a squirt bottle or resealable plastic bags with small holes snipped in the corners.