Preheat the oven to 425 degrees F.
Heat the butter in a large stock pot over medium heat. Add in the onions, carrots, celery and garlic with a big pinch of salt and pepper. Add the thyme and oregano. Cook, stirring often, until the veggies soften, about 5 to 6 minutes.
Stir in the flour to create a roux. Stir well so it coats all the vegetables, then cook for 3 to 4 minutes, stirring often. It should smell fragrant and nutty.
Add in the chicken and broccoli. Slowly stream in 4 cups of the chicken stock while stirring. Doing this slowly will help the mixture thicken.
Bring the mixture to a boil. Reduce it to a simmer and cook for 5 more minutes so it can thicken. Stir in the heavy cream and cook for another few minutes. Turn off the heat and stir in the cheese. Stir until it melts completely.
Taste and season with more salt and pepper as needed. You may need to add a little more salt and pepper here!
Ladle the mixture into a 9x13 inch baking dish. Top with the sheet of puff pastry. Brush the sheet with the egg wash. Use a knife to slice a few slits into the pastry to let the steam escape.
Place the dish in the oven. Bake for 25 to 30 minutes or until the puff pastry is golden and puffed. Remove and let cool slightly before serving!