Halve potatoes lengthways and thinly slice widthways. Place in a bowl, cover with cold water and set aside.
Heat olive oil in a large saucepan over medium heat, add two-thirds of the onion and sauté until soft (4-5 minutes). Add mussels and wine and bring to the boil. Cover and cook until mussels open (3-4 minutes) then drain and, when cool enough to handle, remove mussels from shells and place in a large bowl (discard liquid and shells).
Heat vegetable oil in a large deep saucepan or deep-fryer to 150C. Add garlic, cook until golden (3-4 minutes), then transfer to an absorbent paper-lined plate using a slotted spoon.
Drain potatoes and pat dry with a clean tea towel. Blanch in oil for 10 minutes, then transfer to an absorbent paper-lined plate. Increase heat to 180C, then deep-fry in batches until golden brown and crisp (2-3 minutes). Transfer to a large bowl and season with sea salt and paprika. Add mussels, remaining onion and mint, drizzle with extra-virgin olive oil and lemon juice, and toss to combine. Divide among plates, scatter garlic over and serve immediately with mayonnaise passed separately.