Cut Brie in half horizontally into 2 rounds; set aside.
On lightly floured surface, roll out pastry to 1/8-inch thickness; cut into 12-inch rounds. Place 1 of the cheese rounds in centre. Spread chutney over cheese; top with remaining cheese round. Fold pastry up over cheese, overlapping edge and sealing with some of the egg; brush top with remaining egg. (Make-ahead: Cover and refrigerate for up to 4 hours.)
Place on rimmed baking sheet; bake in 400°F oven until golden and puffed, 25 to 30 minutes. Let stand for 20 minutes. To serve, cut into wedges with serrated knife.