In a small bowl, whisk water and salt together until salt dissolves. Transfer salt water and potatoes to the bowl of a food processor fitted with the metal blade and process until coarsely ground, 10 to 12 pulses, stirring occasionally.
Drain potato mixture in fine-mesh strainer, pressing potatoes with a spoon until dry (liquid should measure about 1½ cups); discard liquid. Transfer potatoes to a microwave safe bowl and microwave, uncovered, until dry and sticky, 8 to 10 minutes, stirring halfway through cooking.
Stir flour, black pepper and cayenne pepper(optional) into potatoes. Spread potato mixture into thin layer over large sheet of aluminum foil and let cool for 10 minutes. Push potatoes to center of foil and place foil and potatoes in 8-inch square baking pan. Push foil into corners and up sides of pan, smoothing it flush to pan. Press potato mixture tightly and evenly into pan. Freeze, uncovered, until firm, about 45 minutes.
Meanwhile, preheat oven to 200 degrees F. Set a wire rack in a rimmed baking sheet; set aside. Heat oil in a saucepan over high heat until 375 degrees. Using foil overhang, lift potatoes from pan and cut into 1¼ by 1-inch pieces (6 cuts in 1 direction and 8 in other). Fry half of potato tots, until golden brown and crisp, 5 to 7 minutes, stirring occasionally once they begin to brown. Drain prepared baking sheet, season to taste and place in oven. Bring oil back to 375 degrees and repeat with remaining potato tots. Serve warm.
If your food processor has a capacity of less than 11 cups, you’ll need to process the potatoes in two batches.
If any large pieces of potato remain after processing, chop them coarsely by hand.
To make handling the uncooked tots easier, use a wet knife blade and wet hands.
To Make Ahead: Cool fried potato tots, transfer to zipper-lock bag, and freeze for up to 1 month. To serve, adjust oven rack to middle position and heat oven to 400 degrees. Place potato tots on rimmed baking sheet and bake until heated through, 12 to 15 minutes.