Cut 1.35kg of whole chicken into wings, thighs, drumsticks, and halved breasts, and chop the carcass into 2in / 5cm pieces with a cleaver.
In a heavy pot over medium-high heat, cook the carcass pieces and wings with 1½ cups of onion in 1in / 2.5cm pieces, stirring now and then, until the chicken begins to brown.
Add ½ cup of water, 2 sprigs of fresh parsley, and 2 sprigs of fresh thyme, then cover and cook over low heat for 20 minutes.
Add 6 cups of hot water, the remaining chicken pieces, and 1 teaspoon of table salt. Simmer, partly covered, for about 20 minutes, until the chicken pieces are just cooked through.
Strain the stock. Shred the meat from the chicken pieces, discarding the skin and bones, and wipe out the pot.
Melt ½ cup of unsalted butter in the pot. Cook 1 cup of diced onion, 1 cup of diced carrots, and 3½ cups of stemmed, quartered cremini mushrooms for about 6 minutes, until just tender but not brown.
Sprinkle in ¾ cup of all-purpose flour and cook, stirring, for 2 minutes. Whisk in 5 cups of the stock, 1 cup at a time, bringing it to a simmer after each addition.
When the gravy thickens, stir in ½ cup of chopped fresh parsley, 1 teaspoon of chopped fresh thyme, and 3 cups of the shredded chicken. Season with salt and pepper, stir in ¼ cup of heavy cream, and return to a simmer.
For the dumplings, whisk 1¾ cups of all-purpose flour, 2 teaspoons of baking powder, 1 teaspoon of kosher salt, and 1 teaspoon of sugar. Stir in 1 large egg whisked with 1 cup of milk.
Stir in ¼ cup of minced fresh herbs (tarragon, chervil, chives, or parsley) and 2 thinly sliced scallions. Drop 1-tablespoon spoonfuls all over the simmering stew. Cover and simmer for 10 minutes, until the dumplings are cooked through.
