Preheat your oven to 425 degrees F. Line 2 baking sheets with parchment paper and set aside.
To a skillet, melt the butter with the brown sugar, about 2-3 minutes. Add the chopped walnuts and stir to coat. Cook for 2-3 minutes more or until the walnuts just begin to darken. Transfer the mixture to one of the baking sheets and spread out to a thin layer. Once cool enough to handle, roughly chop the walnuts. Set aside.
Place the thawed puff pastry sheet on the middle of the second baking sheet. Cut the rind of one side of the brie wheel. Place the brie on the middle of the puff pastry sheet (cut side up). Spread the apricot jam on top and pile the chopped apricots and candied walnuts on top. Taking the corners of the puff pastry, cinch it up and into the center of the brie. Pinch the puff pastry where it breaks open. It's OK if it breaks open a bit during baking. Brush the pastry with the beaten egg and sprinkle of coarse sugar.
Bake the brie for 20-25 minutes or until brown and crispy. Remove from oven and let stand for 5 minutes before serving. Serve with crackers, crispy bread, and sliced apples. Enjoy!