1. Bring all ingredients and 12 cups cold water to a boil in a very large (at least 12-qt.) stockpot. Reduce heat to medium-low and simmer until chicken breasts are cooked through, about 20 minutes.
2. Transfer breasts to a plate (remaining chicken parts are strictly for stock). Let breasts cool slightly, then remove meat and return bones to stock. Shred meat. Let cool, tightly wrap, and chill.
3. Continue to simmer stock, skimming surface occasionally, until reduced by one-third, about 2 hours. Strain chicken stock through a fine-mesh sieve into a large saucepan (or airtight container, if not using right away); discard solids. You should have about 8 cups.

DO AHEAD: Stock can be made 2 days ahead. Let cool; cover and chill. Keep reserved chicken meat chilled.

1. Mix eggs, matzo meal, schmaltz, club soda, and salt in a medium bowl (mixture will resemble wet sand; it will firm up as it rests). Cover and chill at least 2 hours.

DO AHEAD: Mixture can be made 1 day ahead. Keep chilled.

1. Bring chicken stock to a boil in a large saucepan. Add carrots; season with salt. Reduce heat and simmer until carrots are tender, 5–7 minutes. Remove from heat, add reserved breast meat, and cover. Set soup aside.
2. Meanwhile, bring a large pot of well-salted water to a boil. Scoop out 2-tablespoonful portions matzo ball mixture and, using wet hands, gently roll into balls.
3. Add matzo balls to water and reduce heat so water is at a gentle simmer (too much bouncing around will break them up). Cover pot and cook matzo balls until cooked through and starting to sink, 20–25 minutes.

DO AHEAD: Using a slotted spoon, transfer matzo balls to bowls. Ladle soup over, top with dill, and season with pepper.