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Preheat oven to 425 degrees F.
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Whisk 1 1/4 cups broth and flour in a medium bowl; set aside.
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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.
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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.
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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.
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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.
Tips
Tip: To poach chicken: Place 4 small boneless, skinless chicken thighs or 2 small boneless, skinless chicken breasts (12 oz. total) in a medium saucepan; cover with cold water. Bring to a boil over medium-high heat. Reduce heat to low, partially cover, and cook until an instant-read thermometer inserted in the thickest part registers 165 degrees F, 12 to 15 minutes. Transfer to a plate and shred into small pieces. Reserve broth for use in another recipe–it will keep in the fridge for up to 4 days or in the freezer for up to 3 months.
Equipment: 10-inch nonstick oven-safe skillet