Step 1

Preheat oven to 475 degrees, with a rimmed baking sheet on center rack. In a pot, cover potatoes with 2 inches of water; season with salt. Bring to a boil, then reduce heat and simmer until just cooked through, 6 to 8 minutes. Drain; let stand until dry, 10 minutes. Toss with 3 tablespoons oil; season with salt and pepper. Carefully transfer, cut-side down, to baking sheet and roast, flipping once, until crisp, 20 to 25 minutes. Season with salt.

Step 2

Meanwhile, return pot to medium-high heat; swirl in 1 tablespoon butter and remaining oil. Add garlic; cook until golden, 1 to 2 minutes. Stir in tomatoes and wine; season with salt and pepper. Simmer until reduced slightly, 3 minutes. Stir in mussels, cover, and cook until all have opened (discard any that haven't), 3 to 5 minutes. Transfer mussels to a bowl with a slotted spoon; stir remaining butter into broth. Pour sauce over mussels and sprinkle with parsley. Serve with oven fries and mayonnaise.