Preheat the oven to 400 degrees. Line a sheet pan with parchment paper.
Use cheese wire or a sharp knife to cut the brie round in half horizontally. Spoon 2 tablespoons of the cooled compote onto the cut side of one of the brie rounds. Place the other brie round, cut side down, on top of the compote. Reserve the remaining compote for serving.
Lightly dust a clean work surface with flour, and gently roll out the puff pastry dough so that it’s about ¼-inch thick. Place the brie in the center of the pastry, and wrap the dough around the brie round, so that the brie is fully enclosed in the pastry. Pinch the ends together, and trim off any excess dough. (If you like, use the excess dough to cut out leaves or other shapes for decorating the baked brie.) Chill in the refrigerator for 20 minutes.
When you’re ready to bake, place the pastry-wrapped brie on the prepared sheet pan. In a small bowl, beat the egg with 2 teaspoons of water, and brush over the pastry. Bake until the cheese is hot and gooey, and the pastry is golden brown and flaky, about 35 minutes.
Let the baked brie rest for 5 minutes, then transfer to a serving platter. Serve warm alongside the remaining compote, crackers, baguette slices, and fresh fruit.