Finely chop red cabbage in a food processor.

Transfer cabbage to a bowl and add salt and beet liquid.

Finely chop pickled beets and stir into cabbage mixture along with cider vinegar.

Let slaw stand for 30 minutes, stirring occasionally.

Remove skin and fat from duck leg and finely chop them.

Cook duck skin and fat in a skillet over moderately low heat until rendered and crisp.

Transfer duck skin to paper towels to drain, reserving fat in skillet.

Remove duck meat from bone and dice.

Add onion to skillet and cook over moderate heat until browned.

Combine onion, duck meat, and reserved skin in a large bowl.

Peel potatoes and shred using a box grater.

Squeeze potatoes to eliminate excess moisture.

Add potatoes to duck mixture along with egg, pepper, and salt.

Spread bread crumbs on a sheet of wax paper.

Make 12 mounds of potato mixture on crumbs.

Coat mounds with remaining crumbs and flatten into patties.

Heat vegetable oil in a skillet over moderate heat.

Shake off excess crumbs from each potato pancake.

Fry pancakes in batches, turning over once, until golden.

Transfer to paper towels to drain.

Serve immediately with slaw.