Preheat oven to 425 degrees F.
Whisk 1 1/4 cups broth and flour in a medium bowl; set aside.
Heat oil in a 10-inch cast-iron or oven-safe nonstick skillet over medium-high heat. Add mushrooms; cook until browned, 6 to 8 minutes. Reduce heat to medium. Add onion and garlic; cook until tender, 4 to 6 minutes.
Stir in the remaining 1/4 cup broth, scraping up any browned bits. Stir in the reserved broth-flour mixture, peas and carrots, sage, thyme, salt, and pepper. Bring to a boil, stirring constantly. Reduce heat and cook, stirring occasionally, until the vegetables are tender, about 10 minutes. Remove from heat; stir in chicken.
Lay pie crust over the chicken mixture, folding the edges over as needed. Cut four 4-inch slits in the crust to allow steam to escape. Brush with egg white.
Bake until the crust is golden and the filling is bubbly, 20 to 25 minutes. Let cool for 10 minutes before serving. Garnish with thyme sprigs, if desired.