Place potatoes in a large pot. Cover with cold water and add 2 tablespoons salt. Bring to a boil over high heat and cook 5 minutes, until potatoes are just slightly cooked on the outside. The interiors should still be raw.
Drain and return to the pot. Let moisture evaporate, lightly shaking pot, 1 minute. Turn off heat. Add melted butter and place cover on pot. Shake vigorously for 1 minute, until the potatoes are rough on the exterior and lightly smashed. Transfer to a rimmed baking sheet and spread in an even layer. Roast until crispy and deeply golden, flipping after 30 minutes, about 45 minutes. Season to taste with salt and dress with desired topping.
FOR PATATAS BRAVAS:
In a medium pot, warm olive oil over medium-high heat. Add onions and sliced garlic, season to taste with salt and pepper, and sauté until softened and beginning to turn golden, about 8 minutes. Add smoked paprika and red pepper flakes. Cook until fragrant, stirring, one minute more. Add tomatoes and white wine vinegar and continue to cook, breaking up tomatoes with the back of a spoon or potato masher. Bring mixture to a boil and simmer for 15 minutes. Blend until smooth with hand blender, season to taste with salt and pepper, and keep warm until ready to serve.
In a small bowl, whisk to combine grated garlic, lemon juice, and mayonnaise until smooth. Season to taste with salt.
Serve aioli and tomato sauce with crispy roasted potatoes. Garnish with parsley and serve.
In the bowl of a food processor, combine cilantro, parsley, and mint leaves. Pulse until roughly chopped. Add white wine vinegar, olive oil, sugar, and garlic. Continue to pulse until finely chopped and well-combined, scraping down the sides of the bowl as necessary. Season to taste with salt and serve with crispy roasted potatoes.
Place cucumber in a small bowl and add 1 tablespoon salt. Toss to coat and set aside for 5 minutes. Squeeze cucumbers between paper towels to expel liquid. Rinse with water and pat dry. Transfer to a medium bowl. Add yogurt, garlic, and salt to taste. Serve tzatziki with crispy roasted potatoes, garnished with oregano, lemon zest, and a drizzle of olive oil.