Grate the onion on the side of a box grater with the largest holes until you have 1/3 cup. Place in a large bowl and reserve the rest of the onion for another use.
Shred the potatoes in a food processor fitted with a grating disc with medium-sized holes. Remove and discard any large chunks. Immediately and thoroughly toss with the onion.
Line a smaller bowl with cheesecloth and fill with the potato-onion mixture. Twist into a ball and wring out as much of the liquid as possible, reserving the liquid. Let the extracted liquid sit undisturbed for 5 minutes to allow the starch to settle, then pour off the liquid so that only the starch remains.
Whisk the egg into the starch and set aside.
Add the schmaltz to a cast-iron skillet to measure 1/8 inch deep and heat over medium until shimmering. (Don't let it get over 375ºF; that's its smoke point.)
Place the potatoes in a large bowl, add the instant mashed potatoes, salt, and pepper, and combine using your hands. Stir in the egg mixture.
Drop a mounded tablespoon's worth of the potato mixture into the skillet and flatten firmly with the back of a spatula. Fry in batches of 4 until browned, flipping only once, 2 to 3 minutes per side, then remove to a rack set over a pan lined with paper towels. Season immediately with additional salt. Continue to add fat in between batches to maintain 1/8-inch depth.