Melt the butter in a large skillet over medium heat. Add the onion, carrots, celery, mushrooms, and garlic, and cook for 5 minutes, stirring occasionally, until the vegetables soften and release their liquid.
3 tablespoons unsalted butter, ½ cup chopped yellow onion, 1 cup diced carrots, 1 cup diced celery, 1 cup mushrooms, 3 garlic cloves
Stir in the flour, salt, pepper, thyme, and rosemary. Cook the mixture until the flour absorbs the liquid and thickens slightly.
¼ cup all-purpose flour, 1 teaspoon salt, ½ teaspoon fresh ground pepper, 1 teaspoon dried thyme leaves, ½ teaspoon finely chopped dried rosemary
Gradually add the chicken broth and milk, stirring constantly to avoid lumps. Bring the mixture to a simmer and cook for 7-8 minutes, until it thickens into a creamy gravy.
2 cups chicken broth, ½ cup whole milk
Stir in the frozen peas and turkey, then remove from heat. Taste and adjust seasoning with more salt, pepper, thyme, or rosemary if needed. Let the filling cool for 10 minutes.
1 cup frozen peas, 3 cups turkey
Preheat the oven to 350°F (180°C). Transfer the cooled filling to a 9-inch pie dish. Carefully place the pie crust over the filling, trimming any excess dough and crimping the edges to seal. Cut small slits into the top of the crust for steam vents and brush the surface with the egg wash.
1 pie crusts, 1 large egg beaten with 1 tablespoon milk
Bake for 30-35 minutes, or until the crust is golden brown. Check after 20 minutes and cover the edges with foil or a pie crust shield to prevent over-browning. Let the pie cool for about 5 minutes before serving.