FOR THE CRUST
1 1/4 cups all-purpose flour (spooned and leveled), plus more for work surface
1 teaspoon sugar
1/4 teaspoon fine salt
1/2 cup (1 stick) cold unsalted butter, cut into 1/2-inch pieces
3 to 5 tablespoons ice water
FOR THE FILLING
5 tablespoons unsalted butter
1 medium yellow onion, diced small (1 1/2 cups)
4 medium carrots, diced small (2 cups)
1-2 small potatoes
2 garlic cloves, minced
1/2 cup all-purpose flour (spooned and leveled)
4 cups low-sodium chicken broth
1 cup frozen peas
1 cup frozen corn
Coarse salt and ground pepper
3 cups shredded cooked chicken (15 ounces)
Make the crust: In a food processor, pulse together flour, sugar, and salt. Add butter and pulse until mixture resembles coarse meal, with a few pea-size pieces of butter remaining. Sprinkle with 3 tablespoons ice water. Pulse until dough is crumbly but holds together when squeezed (if necessary, add up to 2 tablespoons ice water); do not overmix.
Form dough into a disk, wrap tightly in plastic, and refrigerate until firm, 1 hour or overnight (or freeze, up to 1 month).
Preheat oven to 375 degrees. Make the filling: In a large pot, melt butter over medium-high. Add onion, potatoes and carrots and cook until softened, 8 minutes. Add garlic and cook until fragrant, 30 seconds. Add flour and stir to coat vegetables.
Slowly add broth, whisking constantly until sauce is smooth. Bring to a boil, reduce heat, and simmer until thickened, 5 to 7 minutes. Stir in peas & corn. Season with salt and pepper, then stir in chicken. Pour filling into a 2-quart baking dish.
On a floured work surface, roll out dough to 1/8-inch thickness. Place dough over dish and fold overhang inward while pinching to crimp edge. Cut vents in dough. Place dish on a rimmed baking sheet and bake until crust is golden brown and filling is bubbling around edge, 45 to 50 minutes. Let cool 15 minutes before serving.