Heat oven to 425° F. Remove meat from chicken, discarding the skin and bones. Shred chicken into bite-size pieces and place in a large bowl. Melt 5 tablespoons of butter in a medium saucepan over medium heat. When butter bubbles, whisk in the flour. Cook, whisking constantly, until the mixture turns golden brown, about 3 minutes. Slowly add the broth in a steady stream, whisking constantly to prevent lumps from forming. When all of the broth has been added, increase the heat to medium-high and bring to a boil. Reduce heat to medium-low and simmer, whisking occasionally for 5 minutes, or until thickened slightly. Remove from the heat.
Melt 2 of the remaining tablespoons butter in another large skillet over medium heat. Add onion and cook, stirring, until translucent, about 5 minutes. Season with 1/2 teaspoon salt and 1/4 teaspoon pepper. Add thyme and carrots and cook just until carrots are tender, about 5 more minutes. Transfer vegetables with a slotted spoon to the bowl containing the chicken. Melt remaining butter in the same skillet over medium heat. Add the mushrooms and cook over medium-high heat until mushrooms release their liquid, about 2 minutes. Reduce heat to low, season with 1/2 teaspoon salt and 1/4 teaspoon pepper and simmer until liquid evaporates. Remove from the heat and add to chicken along with the peas, reserved sauce and remaining salt and pepper. Toss well and spoon into a 9-by-13-inch baking dish.
Lightly flour a clean work surface. Roll dough until it is slightly larger than the baking dish and about 1/4 inch thick. Carefully place the dough over the pie filling, tucking the edges underneath. Bake until crust is golden brown, about 20 minutes. Reduce heat to 350° F and bake until the filling starts to bubble at the edges, 20 to 25 minutes more. Cool for about 5 minutes before serving.