Place Brie cheese in freezer until firm enough to cut, about 30 minutes.

Meanwhile, place piecrust on a clean work surface; if necessary, roll crust into a 10 ½-inch round. Fold outer ½-inch of crust under to make a 10-inch round; crimp as desired. Place on a large plate or baking sheet, and refrigerate until ready to use.

Preheat oven to 400°F with rack in upper third position. Heat oil in a 10-inch cast-iron skillet over medium-high until shimmering. Add mushrooms, and spread in an even layer. Cook, undisturbed, until browned on bottoms, about 5 minutes. Stir; cook, stirring occasionally, until tender and browned all over, about 5 minutes. Transfer to a bowl, and set aside. Do not wipe skillet clean.

Reduce heat in skillet to medium; add butter, and stir until melted. Add onion, carrots, and celery; cook, stirring occasionally, until onion is translucent, about 6 minutes. Stir in garlic and thyme; cook, stirring constantly, until fragrant, about 1 minute. Add sherry; cook, stirring constantly, until fragrant and mostly evaporated, about 1 minute. Stir in flour; cook, stirring constantly, until vegetables are evenly coated with flour and no dry spots remain, about 1 minute.

Add broth to skillet, whisking constantly, until broth is cloudy and no clumps of flour remain, about 2 minutes. Increase heat to medium-high, and bring to a simmer; cook, stirring often, until thickened, about 5 minutes. Remove from heat, and whisk in mustard, salt, and pepper. Fold in chicken and reserved mushrooms. Let stand until cooled slightly, about 10 minutes.

Meanwhile, remove Brie from freezer. Cut away rind, and discard. Cut Brie into ¼-inch-thick slices. Arrange Brie slices in an even layer over top of cooled chicken mixture, cutting slices to fit so they don’t overlap. Place chilled crimped crust on top of Brie slices; cut 3 or 4 (1 ½-inch) slits in center to allow steam to escape. Brush crust with beaten egg.

Place skillet on a large rimmed baking sheet; bake until crust is golden brown and sauce is bubbling around edges, 25 to 30 minutes. Remove from oven, and let stand 10 minutes before serving.