Preheat oven to 400 degrees.
In a large skillet set over medium high heat add butter. Once the butter is melted add in the yellow onion, fresh thyme and fresh rosemary. Saute until the onion is soft and golden in color. Add in the garlic and saute just until fragrant, about 30 seconds.
Add in the flour and stir to combine. Lower the heat to medium and cook until the mixture smells toasted and is golden in color.
Slowly whisk in chicken stock until the mixture is smooth and the flour is no longer clumpy. whisk in milk and continue to cook until the mixture is thick and will coat the back of a spoon. This can take anywhere from 10-15 minutes. Season with desired amount of kosher salt and black pepper.
Add in the carrots and continue cooking just until they start to soften, about 5 minutes.
Remove from the heat and stir in shredded turkey, frozen peas, chopped broccoli and parsley. Stir until the frozen broccoli and peas are combined and don't remain in large clumps.
Pour filling of the pot pie into a 2 quart dish and top with lightly rolled out puff pastry. You're just looking for the puff pastry to cover the dish and a little overhang is ok.
Cut 3 slits into the top of the puff pastry. Glaze the top with the beaten egg.
Bake for 25 minutes and then lower the heat to 350 degrees and bake for another 35-40 minutes or until the topping is golden brown and crispy. The filling might bubble a bit around the edges and through the top so it's best to bake on a sheet pan so you don't risk any spilling in your oven.
Let coolfor about 10 minutes before digging in.