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, 1/2 teaspoon smoked paprika, and 1/2 teaspoon 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 with paper towels to cool slightly. Toss in a bowl with spice blend and serve with dipping sauce