Skillet Chicken Pot Pie with Biscuit Topping

For the biscuits: Adjust an oven rack to the upper-middle position and heat the oven to 450 degrees.

Whisk the flour, sugar, baking powder, and salt together in a large bowl. Stir in the cream with a wooden spoon until a dough forms, about 30 seconds. Turn the dough out onto a lightly floured work surface and gather into a ball. Knead the dough briefly until smooth, about 30 seconds.

Pat the dough into a 3/4-inch-thick circle. Cut the biscuits into rounds using a 2 1/2-inch biscuit cutter or cut into 8 wedges using a knife.

Place the biscuits on a parchment paper-lined baking sheet. Bake until golden brown, about 15 minutes. Set aside on a wire rack.

For the filling: While the biscuits bake, pat the chicken dry with paper towels and season with salt and pepper. Melt 2 tablespoons of the butter in a 12-inch skillet over medium heat until the foam subsides. Brown the chicken lightly on both sides, about 5 minutes total. Transfer the chicken to a clean plate.

Add the remaining 2 tablespoons butter to the skillet and return to medium heat until melted. Add the onion, celery, and 1/2 teaspoon salt and cook until the onion is softened, about 5 minutes. Stir in the flour and cook, stirring constantly, until incorporated, about 1 minute.

Stir in the vermouth and cook until evaporated, about 30 seconds. Slowly whisk in the broth, cream, and thyme, and bring to a simmer. Nestle the chicken into the sauce, cover, and cook over medium-low heat until the thickest part of the breast registers 160 degrees on an instant-read thermometer, 8 to 10 minutes.

Transfer the chicken to a plate. Stir the peas and carrots into the sauce and simmer until heated through, about 2 minutes. When the chicken is cool enough to handle, cut or shred it into bite-sized pieces and return it to the skillet. Season the filling with salt and pepper to taste.

For serving: Place the cooked biscuits on top of the skillet or individual servings.