Position rack in upper third of oven; preheat broiler.
Heat 1 tablespoon oil in a large ovenproof skillet over medium heat. Add celery and shallot; cook, stirring, until the shallot begins to brown, about 2 minutes. Add bread and poultry seasoning; cook, stirring, until the celery has softened and the bread begins to crisp, 2 to 3 minutes. Transfer the stuffing to a medium bowl and toss with 1/4 cup broth; set aside. Wipe out the pan.
Toss chicken with 2 tablespoons flour, pepper and salt in a large bowl. Whisk the remaining 1 1/4 cups broth with the remaining 2 tablespoons flour in a small bowl and set aside.
Add the remaining 1 tablespoon oil to the pan and heat over medium heat. Add the chicken (shaking off any excess flour) and mushrooms; cook, stirring, until the chicken is cooked through, about 8 minutes.
Increase heat to medium-high; add wine (or sherry) and cook, scraping up any browned bits with a wooden spoon, until almost evaporated, about 2 minutes. Stir in the reserved broth-flour mixture and cook, stirring, until thickened, about 2 minutes more. Spoon the reserved stuffing over the chicken mixture. Transfer the pan to the oven and broil until the stuffing begins to crisp, about 4 minutes.
TIPS & NOTES:.
Shopping tip: Look for “salt-free” poultry seasoning near other spice mixes. If you can only find poultry seasoning with salt, reduce the salt to 1/8 teaspoon.
Tip: To trim boneless, skinless chicken thighs, we like to use kitchen shears to snip the fat away from the meat. After trimming, you'll have four 4-ounce portions.