Step 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.

Step 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.

Step 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.

Step 4

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

Step 5

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.

Step 6

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

Step 7

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.

Step 8

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.

Step 9

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.

Step 10

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