Preheat oven to 400°F. Line a large rimmed baking sheet using parchment paper. Roll out piecrust to a 13-inch round. Place on the prepared baking sheet. Chill piecrust round while preparing filling.
Melt butter in a large skillet over medium-high heat. Add onion and garlic. Cook, stirring often, until just tender, about 4 minutes. Add flour, and cook, stirring constantly, 1 minute. Stir in cream. Bring to a simmer, stirring constantly, until thickened slightly, about 1 minute. Remove from heat. Stir in chicken, frozen vegetables, thyme, salt, pepper, and ½ cup of the cheese (makes 3 ½ cups).
Spoon chicken mixture into the center of the piecrust round, leaving a 1 ½-inch border. Fold edges up and over filling, overlapping folded edges as needed, and pinching to seal folds to help the crust maintain its shape. Brush piecrust with beaten egg. Sprinkle crust with remaining ½ cup cheese and additional black pepper.
Bake in preheated oven until piecrust is browned and filling is bubbly, about 30 minutes. Remove from oven, and let cool 10 minutes before serving. Garnish with additional thyme.
