Using a box grater or food processor fitted with the fine grater blade, grate the onions; transfer to a colander set over the sink to drain and discard the liquid, then transfer to a large bowl.

 

Repeat with the potatoes, then drain them as well, pressing to extract as much liquid as possible. Add the drained potatoes to the bowl along with the onions, then stir in the eggs, salt, and black pepper until thoroughly combined.

 

Mix in ¼ cup of the flour, then slowly add more flour a tablespoon at a time until the mixture has the consistency of thick oatmeal (you may not need it all).

 

To a deep, heavy-bottomed skillet, add oil to a depth of 1 inch. Cook over medium-high heat until the oil is hot and shimmering (about 325° on a deep-fry thermometer). Line a large rimmed baking sheet with paper towels and set it by the stove.

 

When the oil is hot, drop in a scant ¼ cup of the mixture. Press down slightly with the back of a fork to form an even latke, about ¾ inch thick. Cook until the bottom and edges of the latke turn a deep golden brown, 2–3 minutes. Using a fork, metal spatula, or metal slotted spoon, flip the latke and continue cooking until evenly browned, 2–3 minutes more.

 

Transfer to the lined baking sheet while you continue cooking the rest of the latkes. Serve hot or warm, with applesauce or sour cream on the side.