Combine the flour and salt in a bowl. Cut in the cold butter until the mixture resembles coarse crumbs with pea-sized pieces. Drizzle in the ice water, tossing until the dough just holds together. Shape into a disk, wrap, and chill at least 30 minutes.

Melt the 3 tbsp butter in a 10-inch oven-safe skillet over medium heat. Cook the onion until soft, 5 to 7 minutes, then add garlic. Stir in the flour and cook 1 minute. Slowly whisk in the broth and milk, bring to a simmer, and cook until thickened, 3 to 5 minutes. Stir in mustard, salt, pepper, thyme, chicken, and frozen vegetables. Remove from heat.

Preheat oven to 400°F. Roll the dough on a floured surface into a 12-inch circle. Drape it over the filling, crimp the edges, cut slits, and brush with beaten egg.

Bake until deep golden and the filling bubbles, 30 to 35 minutes.

Let rest for 10 minutes, then slice and serve.