Melt butter and olive oil in a large skillet over medium heat.
Add the onions and evenly sprinkle with salt. Cook over medium heat, stirring occasionally, until deep brown and caramelized about 45 minutes.
At about 25 minutes into cooking, the onions will begin to stick and leave browned bits in the bottom of the pan. A few tablespoons at a time, deglaze the pan with water, scraping the browned bits from the pan, and repeat as needed.
Once the onions are caramelized, remove from heat and stir in thyme, garlic, and black pepper. Cool for 10 minutes.
Preheat oven to 400 degrees and line a baking sheet with parchment.
Unroll the sheet of puff pastry on a lightly floured surface and roll to a 10-inch square.
Place puff pastry on a parchment lined baking sheet. Place the wheel of brie into the center of the pastry dough and top with caramelized onions.
Whisk egg with 1 tablespoon of water in a small bowl.
Fold the dough edges up and around the brie and onions, brushing with the egg wash between any part of the dough folded on top of itself.
Once the dough is folded around the brie, brush all over with the egg wash. Discard any excess.
Bake until the puff pastry is evenly browned, about 45 minutes.
Allow to cool for 10 minutes, transer to a serving platter and serve with additional crackers.