Place the potatoes in a medium saucepan and cover with cold water. Bring to a boil and cook for 12 minutes, or until the potatoes are tender. Drain well. Transfer to a large bowl and mash with a potato masher or handheld mixer. Cool to room temperature.
In a food processor, combine the fish, oat bran, egg, lemon juice, paprika, salt, and pepper. Pulse until the fish is finely ground.
Transfer the mixture to the bowl with the potatoes. Add the scallion greens and mix well to combine. Using a ½-cup measuring cup, scoop the mixture and shape into 8 fish cakes. Transfer to a platter and refrigerate for 30 minutes.
In a large nonstick or cast-iron skillet, heat 2 teaspoons of the oil over medium heat. Add half the fish cakes and cook for 3 minutes, or until browned. Flip the fish cakes over and cook for 3 minutes, or until browned. Repeat with the remaining 2 teaspoons oil and the 4 remaining fish cakes.
Serve 2 fish cakes per person, with lemon wedges, if desired.
Note: If you have gluten issues, choose gluten-free brands of oat bran and paprika.