Preheat your oven to 425°F (220°C) to ensure it's hot when you're ready to bake. Line a baking sheet with parchment paper and lay the thawed puff pastry flat on top, smoothing out any creases. This will make it easier to assemble and wrap the brie.
Place the wheel of brie cheese in the center of the puff pastry. Using a sharp knife, carefully score the top of the brie in a crosshatch pattern—this allows the flavors to infuse. Spread the fig jam evenly over the scored surface, drizzle the honey on top, and then sprinkle with fresh thyme leaves and a pinch of black pepper. I like to be generous with thyme for extra aroma.
Layer the slices of prosciutto over the topped brie, wrapping them around the wheel to completely encase it. Next, gently fold each corner of the puff pastry up and over the brie, pressing the seams and edges to ensure it's sealed all around. This will keep the cheese and fillings contained as they melt during baking.
Brush the surface of the wrapped pastry with the whisked egg for a glossy, even finish. Sprinkle a little coarse sugar on top for a delightful crunch and hint of sweetness. Bake the prepared brie in your preheated oven for 20-25 minutes, or until the pastry is deep golden brown and crispy. I always keep an eye on the pastry during the last few minutes—the sugar can caramelize quickly.
Let the baked brie rest for a few minutes after removing it from the oven; this makes it easier to slice and serve. Present it warm, accompanied by your favorite bread or crackers. For an extra touch, I sometimes drizzle a bit more honey on top before serving.