Step 1Cut up the chicken: Remove the giblets, saving the neck (discard the remaining giblets, or save for another use). Using a pair of kitchen shears, cut along the back of the chicken, removing the backbone. Cut or break the backbone into thirds (this will help to flavor and thicken the broth). With a sturdy French knife or cleaver, halve the chicken lengthwise down the breast. Cut each chicken half into 4 pieces, separating the leg and thigh, and halving the breast crosswise (the wing can remain attached to the breast or separated). Sprinkle the chicken pieces (including the neck and back) with 1 1/2 teaspoons salt and several grinds of pepper, evenly seasoning the pieces.
Step 2Heat a large, sturdy stockpot over high heat. When hot, add the olive oil and enough chicken to fit comfortably in a single layer. Brown the chicken on all sides, about 15 minutes (this will probably need to be done in 2 batches). Remove the chicken to a bowl and repeat until all the chicken is browned.
Step 3Reduce the heat to medium-high. To the fat in the pot, add the chopped onion, carrots, celery and leek, cooking until the vegetables just begin to color, 8 to 10 minutes, stirring frequently.
Step 4Stir in the white wine and cook, scraping any flavoring from the bottom of the pan. Continue to cook until the wine is almost evaporated, 2 to 3 minutes.
Step 5Add the garlic, 3 sprigs parsley, thyme and bay leaf to the pot, and add back the chicken.
Step 6Pour in 10 cups water (this should more than cover the chicken), loosely cover the pot and bring the liquid to a boil. Reduce the heat to a gentle simmer and cook, loosely covered, until the chicken is very tender, 45 minutes to an hour. Periodically skim the fat that forms on top of the broth as the chicken cooks.
Step 7When the chicken is tender, remove the pieces to a large plate or baking dish until cool enough to handle. Strain the chicken broth into a separate 3-quart pot, discarding the vegetables and herbs. You should have about 10 cups of broth. Skim any remaining fat from the broth, and season to taste.
Step 8Remove the skin from the chicken pieces and peel the meat from the bones. Shred the meat into bite-sized pieces; you should have about 6 cups of chicken. Place the meat in a bowl and set aside while you make the dumpling batter.
Step 9To make the dumplings: In a medium bowl, whisk together the cornmeal and one-third cup boiling water. Cover the bowl with plastic wrap and set aside until the mixture cools, about 15 minutes. Stir in the flour, 1 teaspoon salt and the baking powder, breaking up any cornmeal clumps with your fingers.
Step 10In a small bowl, whisk together the egg and 2 tablespoons milk. Pour the milk mixture into the medium bowl, and drizzle over the lemon juice. Stir to combine, then gently fold in the herbs until evenly distributed. This should form a thick batter (it should have the consistency of thick cement, sticky yet spoonable). Add more milk if needed to thin the batter, 1 tablespoon at a time. (You may not use all of the milk.)
Step 11Bring the broth to a gentle simmer on the stove. Spoon 1-inch balls of the batter (the dumplings will expand as they cook) into the simmering broth; this makes about 20 dumplings. The dumplings will sink at first but will soon float; continue to simmer, loosely covered, until they are just cooked through, about 15 to 20 minutes. Strain the dumplings onto a large plate or baking dish.
Step 12Add the shredded chicken back to the broth. Serve immediately, adding dumplings back to each serving.