Put the potatoes in a pot that holds them in a single layer and cover with cold water. Bring to a boil over high heat and cook until the potatoes are tender but still hold their shape, about 30 minutes. Drain and let cool. When the potatoes are cool enough to handle, remove their skins.
Pour several inches of vegetable oil into a large heavy pot and bring to 350 degrees F over high heat.
While the oil heats, cut the potatoes into cubes.
Fry the potatoes until golden, 5 to 7 minutes.
Meanwhile, mash the garlic and 1 tablespoon flaky sea salt with a mortar and pestle until a paste forms. Stem and roughly chop the chiles and add them to the mortar; continue mashing. Add the lemon juice, cilantro, and olive oil and mix well. Taste for seasoning and add more salt if necessary.
Remove the potatoes to a heatproof mixing bowl with a slotted spoon.
Pour the dressing over the potatoes and toss to combine. Transfer to a serving dish and sprinkle with paprika, chopped chiles and cilantro.