Place the chicken pieces in a large Pot. Cover with 12 cups of cold water. Bring to a boil over high heat, then immediately skim off the grey foam that rises to the surface using a Ladle or Wooden Spoon. This step is important for a clear, clean-tasting broth — skim for about 5 minutes until the foam subsides.

Add the 3 zanahorias peladas, cortadas en trozos grandes, 3 tallos de apio cortados en trozos grandes, con hojas incluidas, 1 cebolla amarilla grande cortada por la mitad, con piel para color dorado, 1 chirivía pelada, cortada en trozos grandes (if using), 6 dientes de ajo machacados, con piel, 10 ramitas de perejil de hoja plana fresco, 6 ramitas de eneldo fresco más para servir, 1 cucharadita de granos de pimienta negra enteros, and 2 tsp 2 cucharaditas de sal kosher más. Reduce the heat to low so the broth is at a very gentle simmer — you should see only occasional lazy bubbles. A hard boil makes cloudy broth. Simmer uncovered for 2 hours.

After 2 hours, use Tongs to remove the chicken pieces to a Cutting Board. Let cool until safe to handle, then pull the meat from the bones and shred it. Discard the skin and bones. Set the shredded chicken aside for serving.

Strain the broth through a Sieve into a clean Pot, pressing lightly on the vegetables. Discard the spent vegetables and herbs. Taste the broth and season generously with 2 cucharaditas de sal kosher más — a well-seasoned broth makes all the difference. Skim any excess fat from the surface, or reserve it as schmaltz for the matzo balls.

In a medium Mixing Bowl, whisk together the 4 huevos grandes, batidos, schmaltz (or Oil / 2 cucharadas de mantequilla o schmaltz (grasa de pollo), derretida, para las bolas de matzo), and ¼ cup of the warm broth until well combined. Add the 1 taza de harina de matzo, 2 cucharaditas de sal kosher más, 0.25 cucharadita de polvo de hornear el secreto para bolas de matzo extra esponjosas, 1 cucharadita de granos de pimienta negra enteros, and 6 ramitas de eneldo fresco más para servir (if using). Stir with a fork until just combined — do not overmix. The batter will look loose and wet, which is correct.

Cover the Mixing Bowl and refrigerate the matzo ball batter for at least 30 minutes, and up to 2 hours.

Bring a large Pot of salted water to a boil. (Cook the matzo balls in water, not broth — this keeps your broth clear and golden.) Wet your hands with cold water to prevent sticking. Gently scoop about 2 tablespoons of batter per ball and roll into a loose sphere — don't pack or compress them. You should get about 12 balls.

Carefully lower the matzo balls into the boiling water one at a time. Reduce the heat to a steady simmer, cover the Pot, and cook for 35–40 minutes. Do not lift the lid during the first 30 minutes — the steam is what makes them puff up and stay fluffy. They are done when they have doubled in size and are cooked through (cut one open to check — no dense center).

While the matzo balls cook, bring the strained broth back to a gentle simmer in its Pot. Add the fresh 3 zanahorias peladas, cortadas en trozos grandes and sliced 3 tallos de apio cortados en trozos grandes, con hojas incluidas and simmer for 10–12 minutes until just tender. Add the shredded chicken back to the broth and warm through.

Use a Slotted Spoon to transfer 2 matzo balls into each bowl. Ladle the hot broth with chicken, 3 zanahorias peladas, cortadas en trozos grandes, and 3 tallos de apio cortados en trozos grandes, con hojas incluidas over the top. Garnish generously with fresh 6 ramitas de eneldo fresco más para servir and 10 ramitas de perejil de hoja plana fresco. Serve immediately.