Line a large plate with parchment. In a cup, combine the butter, 1 tbsp. scallion greens, the paprika and cayenne. Dollop 12 mounds of the butter onto the parchment. Freeze.
In a medium saucepan, combine the potatoes and enough cold water to cover by 1 inch. Salt the water and bring to a boil; lower the heat and simmer until the potatoes are fork-tender, 15 to 20 minutes. Drain the potatoes, return to the pot and dry over low heat for 2 minutes. Press the potatoes through a ricer into a bowl (or mash well). Stir in the remaining scallions, 1/2 tsp. salt and 1/4 tsp. pepper; let cool. Mix 1 egg yolk into the potato mixture. In a medium bowl, beat the remaining whole egg and egg white. Place the cracker crumbs in another medium bowl.
Divide the potato mixture into 12 portions. Press a frozen butter mound into the center of each and gently enclose in a ball. Coat the potato balls in the crumbs, then in the beaten egg, then again in the crumbs. Transfer to a plate. Cover loosely with a piece of parchment and refrigerate until ready to fry.
In a large saucepan, heat the oil over medium-high heat until it registers 350 degrees on a deep-fry thermometer, about 5 minutes. Add 6 potato balls and fry, turning once with a slotted spoon, until golden and crisp, 2 to 3 minutes. Transfer to paper towels to drain. Allow the oil to return to 350 degrees and repeat with the remaining potato balls.