Preheat the oven to 400°F. Line a baking sheet with parchment paper.
In a large skillet, heat the olive oil over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes. Stir in the minced garlic and cook for an additional minute.
Add the shredded chicken, chicken broth, heavy cream, thyme, salt, and pepper to the skillet. Bring to a simmer and cook for about 5-7 minutes until the mixture thickens slightly. Stir in the frozen peas and remove from heat.
On a lightly floured surface, roll out the puff pastry sheets to about 1/8 inch thick. Cut out circles using a 4-inch round cutter. Cut smaller circles (about 2 inches) from the center of half of the larger circles to create a ring.
Place the full circles on the prepared baking sheet. Brush the edges with the beaten egg. Place the rings on top of the full circles and brush the tops with more egg wash.
Bake in the preheated oven for 15-20 minutes or until golden brown and puffed.
Once baked, remove from the oven and let cool slightly. Carefully spoon the creamy chicken mixture into the center of each vol-au-vent. Serve warm.