Line a 9- x 12-inch jelly roll pan with foil. Tear 2 (15- x 15-inch) sheets of parchment paper and 2 (15- x 15-inch) sheets of foil. Preheat oven to 350°F.
In a food processor, pulse onions, carrots, and brisket scraps (if using) until very fine.
Heat a sauté pan, with beef or chicken fat or evoo, over medium heat. Add onions, carrots, and brisket (if using) and cook until golden, stirring occasionally.
Add garlic and paprika and continue cooking for another 3 minutes.
Transfer mixture, being sure to scrape all the fat, back to food processor and with machine running add matzo meal until the mixture forms a thick dough that holds together when pressed. Add salt and pepper.
Divide dough in half and form 2 (9-inch) logs on each sheet of parchment. Roll parchment around the log and twist ends to seal. Wrap each log in a sheet of foil.
Roast kishke for 1 hour. Unwrap and continue cooking for an additional 15 minutes until browned and crispy.
Slice and serve or add to cholent.