Put potatoes into a saucepan, and add enough cold water to cover. Bring to a boil, and add 1 tablespoon salt. Reduce heat, and simmer until tender, about 30 minutes. Drain, and let cool slightly. While potatoes are warm, peel and pass through a ricer or a food mill.
Heat olive oil in a skillet over medium heat. Add onion, and cook until golden, about 15 minutes. Add onion to potatoes. Stir in 2 eggs, salt, and pepper. Let cool.
Put flour, remaining 2 eggs, and panko into 3 separate dishes. Lightly beat eggs. For each croquette, scoop out 2 tablespoons potato mixture, then shape into an oval. Dredge in flour, coat with egg, then roll in panko. Place on a rimmed baking sheet. Repeat with remaining potato mixture. Cover, and refrigerate for at least 6 hours (or overnight).
In a large cast-iron skillet or a medium saucepan, heat vegetable oil to 350. Fry croquettes in batches (do not overcrowd skillet) until golden brown, about 4 minutes. Turn croquettes occasionally as they cook to brown evenly on all sides. Drain on paper towels. Season with salt. Serve with remoulade.