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In a large, deep pot set over medium-high heat, melt together olive oil and butter. Generously season chicken pieces with salt and pepper. Brown the chicken on both sides, working in batches so as not to crowd the pot (about 4 to 6 minutes per side). Transfer chicken to a plate and set aside.
Add the celery, carrots, and garlic to the pot. Cook the vegetables in the drippings until they begin to soften, stirring and adjusting the heat as necessary, about 5 to 7 minutes. Add the browned chicken pieces, thyme, bay leaves, garlic powder, onion powder, and chicken broth. Bring to a simmer and cook until the chicken is cooked through, about 25 to 30 minutes. Remove and discard the thyme stems and bay leaves. Transfer the cooked chicken to a plate. Remove and discard the skin and bones, shred the chicken, and return to the pot.
Ladle 2 cups of the cooking liquid into a large bowl or quart-sized measuring cup. Whisk in 1/2 cup of the flour until smooth. Continuously whisk the contents of the pot while slowly pouring in the flour mixture. Simmer and whisk until slightly thickened, about 8 to 10 minutes.
To make the dumplings, in a medium bowl, whisk together the remaining flour, cornmeal, baking powder, baking soda, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Stir in the melted butter, buttermilk, and minced parsley until just combined. With heat reduced to low, drop rustic, tablespoon-sized balls into the simmering broth. Gently stir dumplings one time and then simmer, covered, until the dumplings are firm, about 12 to 15 minutes. Stir in sherry and season the dish with additional salt and pepper, to taste. Serve immediately, sprinkled with additional minced fresh parsley.