Preheat your oven to 400°F (200°C) and line a baking sheet with parchment paper.

Roll out the puff pastry sheet on a lightly floured surface. Using a round cookie cutter (about 3 inches in diameter), cut out circles from the pastry.

Take a smaller cookie cutter (about 1.5-2 inches) and press it into the center of each circle, cutting only halfway through to create the vol-au-vent shell shape. Place the pastry circles on the prepared baking sheet.

Brush the tops of the pastry circles with beaten egg wash, being careful not to let it drip down the sides. Bake for 15-18 minutes until golden brown and puffed. Remove and let cool slightly, then press down the centers.

While the pastry bakes, heat butter and olive oil in a large skillet over medium heat. Add the sliced onions and cook, stirring occasionally, for 25-30 minutes until deeply caramelized and golden brown.

Add the balsamic vinegar and brown sugar to the onions. Continue cooking for another 5-7 minutes until the liquid has reduced and the onions are glossy. Season with salt and pepper to taste.

Toast the walnuts in a dry skillet over medium heat for 5-7 minutes, stirring frequently, until fragrant and lightly browned. Roughly chop them once cooled.

To assemble, spoon caramelized balsamic onions into each pastry shell. Top with crumbled blue cheese and toasted walnuts.

Garnish with fresh thyme or parsley leaves. Drizzle with balsamic glaze if desired. Serve immediately while warm and crispy.