1 pound mixed fish head, skeleton, and skin
1 medium onion, sliced
1 carrot, peeled and sliced into rounds
1 large rib celery, cut crosswise into several pieces
1 teaspoon salt
1/4 teaspoon ground white pepper
2 pounds fish fillets (a combination of two or more of the following: whitefish, carp, perch, yellow pike; see headnote)
1 medium onion, coarsely chopped
1 carrot, peeled and coarsely chopped
1 rib celery, coarsely chopped
2 egg yolks
1 1/2 teaspoons salt
1/2 teaspoon ground white pepper
1/2 cup seltzer
Carrot rounds, for garnish
Sliced onion, for garnish
Horseradish, for accompaniment (see below)
To prepare the broth, place the fish head, skeleton, and skin in a large pot, preferably enameled cast iron. Add the onion, carrot, celery, enough cold water to cover the solids, and the salt and pepper. Heat to boiling over high heat, decrease the heat to medium, and simmer for 30 minutes. Remove from the heat and reserve.
Meanwhile, prepare the fish. Examine the fillets carefully for bones, removing any with tweezers. Cut the fish into 1-inch chunks and place half of them in the bowl of a food processor fitted with the metal blade. Pulse until finely ground.
Transfer the ground fish to the bowl of a stand mixer fitted with the paddle. Grind the remaining fish in the processor and combine with the first batch in the mixer bowl.
Without washing the food processor bowl, fill it with the chopped onion, carrot, and celery. Process until very, very finely chopped. Add to the ground fish, along with the egg yolks, salt, and pepper.
With the mixer on low speed, mix well, scraping the bowl frequently, and gradually blend in the seltzer. Let the machine work the fish mixture for 5 minutes.
Check the seasoning as follows so you don’t taste raw fish: Form a small ball of fish, place it on a small plate with 1 or 2 spoonfuls of broth, and cover loosely with plastic. Microwave on high until cooked. If the mixture is not to your taste, adjust the salt and pepper accordingly.
To cook the fish mixture in the microwave, measure about 1/2 cup of clear broth into a 9-inch glass pie plate or other shallow microwave-safe dish.
With wet hands and a wet rubber spatula, shape the fish mixture into oval cakes, using 1/2 cup of fish mixture for each one. Set about 4 in the 9-inch plate, cover loosely with plastic wrap, and place in the microwave oven. Cook at 50 percent power for 10 to 11 minutes, or longer. If there is no turntable, rotate the dish a third around twice during cooking. Test for doneness by cutting one fish cake in half; when cooked, the fish mixture is firm and no longer translucent. (Since every microwave oven is slightly different, your first batch may have to be somewhat experimental.) Repeat with the remaining fish cakes.
Remove from the microwave, let cool to near room temperature, then refrigerate, tightly covered, to chill thoroughly. Serve each with a carrot round, fresh sliced onion, or horseradish, or all three.
To cook conventionally, have the broth at a simmer. Shape the fish cakes as above, then ease them into the simmering broth on top of the bones and skins. Cover the pot and simmer gently for about 40 minutes.
When done, remove the fish cakes. Strain the broth. Arrange the fish in a shallow baking dish, top each cake with a slice of carrot salvaged from the broth, and pour over the broth (yoich, in Yiddish). This will encase the gefilte fish in jelled broth. Or pack the fish and yoich separately. The jelled broth can be cut into cubes to garnish the fish, but it is traditional to serve each cake of gefilte fish topped with a round of carrot, which also symbolizes prosperity.