Preheat the oven to 400 degrees F.
Remove the puff pastry from the packaging and allow it to thaw somewhat. It shouldn't be super soft, just soft enough to unroll and shape without tearing. If you take a long time to prepare the filling, you can pop the pastry into the fridge while you work.
Clean the mushrooms and remove the stems. Chop them into big pieces. Set aside.
Dice the onion and set aside.
Mince the garlic. Set aside.
Chop the herbs, removing the stems from the thyme and rosemary.
Pat the chicken breasts dry with a paper towel. Lightly sprinkle with salt and pepper on each side.
Cut the chicken into bite-sized pieces.
Heat one tablespoon of olive oil in a large skillet over medium-high heat.
Add the chicken and cook until golden brown, turning the meat over a few times so they cook evenly. The meat should be almost, but not quite, cooked through, because you will be putting it in the oven. This should take 7-10 minutes depending on the the thickness of the meat.
Remove the chicken from the skillet and set aside.
Add 2 tbsp of butter and the mushrooms. Cook until the mushrooms begin to soften, 8-10 minutes. Remove the mushrooms to a large bowl and set aside.
Place 2 more tbsp of butter in the skillet and add the onions. Cook until soft and translucent, about 3-4 minutes.
Add the garlic and herbs to the onions and cook until fragrant, about one minute.
Remove the onions, garlic, and herbs, adding them to the mushrooms. Set aside.
Measure out the chicken broth and set aside.
Melt 2 tbsp butter in the skillet over medium heat.
Add 2 tbsp of flour to the butter and whisk continually until completely combined.
Slowly pour in the chicken broth while stirring with the whisk.
Pour in the heavy cream in a steady stream, whisking continually.
Stir the sauce, scraping up the browned bits from the bottom of the pan, until thickened, 2-3 minutes.
Return the chicken, mushrooms, and onion mixture to the skillet. Toss with the sauce and allow to combine over low heat. Lightly salt and pepper to taste.
Tear a sheet of parchment paper slightly larger than the pie dish.
Roll out one sheet of puff pastry on the parchment paper. You can add a little bit of flour to keep it from sticking, but the parchment paper should be sufficient. You should roll the dough to be about 1 1/2 inches larger than the pie dish.
Invert the pie dish and cut out a circle of dough about one inch larger than the dish.
Repeat with the other sheet of puff pastry, making the circle of dough a bit larger than the first circle.
Beat together one egg and one tbsp of water.
Place one circle of dough, ON THE PARCHMENT PAPER, into the pie dish and prick the bottom of the dough with a fork.
Pour the chicken and mushroom filling onto the dough in the pie dish.
Top with the second, slightly larger circle of puff pastry, folding the top edge under the lower edge and firmly pressing together. I like to crimp it like a pie but you can also use a fork.
Use a pastry brush to coat the dough in the egg wash.
Cut slits in the crust to allow steam to escape.
Place the pie on a rimmed baking sheet and bake in the preheated oven for 25 minutes or until the crust is golden.
Allow to cool for at least 15 minutes before cutting.