Combine garlic, 1/2 teaspoon smoked paprika, and a pinch of salt in a mortar and pestle or food processor; process until smooth. Mix in mayonnaise, sherry vinegar, tomato paste, 1/4 teaspoon chipotle chile powder, and cayenne pepper until combined. Cover and refrigerate dipping sauce.
Mix 1 tablespoon salt, black pepper, 1/2teaspoon smoked paprika, and 1/2teaspoon chipotle chile powder together in a small bowl. Set spice blend aside.
Pour water into a large saucepan and stir 1 tablespoon salt, 1 teaspoon smoked paprika, cumin, and bay leaves into water. Bring to a boil, reduce heat to low, and cover; simmer for 10 minutes.
Stir potatoes into water-paprika mixture, bring to a boil, and simmer until potatoes are tender but not fully cooked, 4 to 5 minutes. The tip of a paring knife should easily insert into the center of a potato cube. Drain potatoes and transfer to a wire rack to cool completely, about 45 minutes.
Heat vegetable oil in a large cast iron skillet or Dutch oven to 350 degrees F (175 degrees C). Add cooled potato cubes and cook, stirring frequently, until golden brown, 5 to 6 minutes. Transfer potatoes to a baking sheet lined withpaper towels to cool slightly. Toss in a bowl with spice blend and serve with dipping sauce.
The nutrition data for this recipe includes the full amount of the salt used in the cooking liquid and spice mix. The actual amount of the salt consumed will vary.
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