Heat the oil in a medium-size saucepan over medium heat. Add the onion and saute, stirring occasionally, until softened, about 5 – 7 minutes. Add the garlic and saute until fragrant, about 2 minutes more. Stir in the piment and fry for another minute. Add the canned tomatoes, vinegar, sugar, and 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 the water. Using an immersion blender, carefully puree the mixture until completely smooth. If you find the sauce to be too thick, you can add more water one tablespoon at a time. Taste and adjust the seasoning as necessary. Set aside while you prepare the potatoes.
Peel the potatoes and cut into approximate 1-inch cubes. Transfer the potatoes to a medium stockpot and fill with just enough water to cover them. Add 1 teaspoon of the salt to the water and bring to a boil over high heat. Once boiling, reduce the heat to medium and gently simmer the potatoes until they are just cooked through, and can easily be pierced with a sharp knife, about 10 – 15 minutes. (Be careful not to overcook your potatoes as they will start to disintegrate and lose their shape). Drain the potatoes and transfer to a serving dish. Allow to cool slightly.
While the potatoes cool, whisk together the mayonnaise, lemon juice, olive oil, and water in a small bowl until thoroughly combined. Season to taste with salt. (Alternatively, if you have mayonnaise out of a squeeze bottle, 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 drizzling of the mayonnaise mixture over the potatoes. Sprinkle with the chopped spring onion greens. Serve Basque-style with lots of toothpicks.