Working over a bowl, grate some of the onion, followed by some of the potatoes, on the large-hole side of a box grater. Repeat until all the vegetables are used up.
Sprinkle mixture with salt and transfer it to a sieve set over a bowl. Squeeze out as much liquid as possible from mixture, allowing it to collect in bottom of bowl. Transfer mixture to another bowl and cover surface with plastic wrap; set aside. Set reserved potato liquid aside to let the milky white starch settle. Pour off liquid from starch. Transfer starch to mixture along with the chives, matzo, eggs, and salt and pepper. Gently mix.
Pour enough oil into a skillet that it reaches a depth of 1⁄4"; heat over medium-high heat. Working in small batches, form mixture into balls, using about 1⁄4 cup of the mixture for each, and place them in the oil. Flatten each ball gently with a spatula to form 3"-4" pancakes. Fry, turning once, until golden brown, crisp, and cooked through, about 8 minutes. Transfer the pancakes to a paper towel-lined plate to drain. Serve the potato pancakes with sour cream or applesauce.