For the salsa brava: Heat the oil in a medium saucepan over medium heat. Add the onions and saute, stirring occasionally, until softened, 5 to 7 minutes. Add the garlic and saute until fragrant, about 2 minutes more. Stir in the piment d'espelette and fry for another minute. Add the sugar, vinegar, canned tomatoes and 1/2 teaspoon salt. Reduce the heat to low and simmer, stirring occasionally, until the tomatoes have broken down and the flavors have had a chance to meld, about 10 minutes.
Remove from the heat and stir in 1/4 cup water. Carefully puree the mixture until completely smooth, using an immersion blender or in batches in a blender. If the sauce is too thick, add water 1 tablespoon at a time. Taste and adjust the seasoning.
For the potatoes: Put the potatoes in a medium stockpot and fill with just enough water to cover. Add 1 teaspoon salt and bring to a boil over high heat. Reduce the heat to medium and simmer gently until the potatoes are just cooked through and can be easily pierced with a sharp knife, 10 to 15 minutes. (Be careful not to overcook the potatoes as they will start to disintegrate and lose their shape.) Drain the potatoes and transfer to a serving dish. Let cool slightly.
For the mayonnaise: While the potatoes cool, whisk together the mayonnaise, olive oil, lemon juice and 1 teaspoon water in a small bowl until thoroughly combined. Season to taste with salt. (Alternatively, you can skip this step and garnish the potatoes with just a squeeze of plain mayonnaise).
To serve, spoon the salsa brava and a generous drizzle of the mayonnaise over the potatoes. Sprinkle with the spring onions and serve Basque-style with a lot of toothpicks.