Place a 9-inch springform pan on a baking sheet lined with parchment paper.
In a large bowl, combine 3 3/4 cups of the flour and 3/4 teaspoon of the salt using your hands or a pastry cutter.
Add 3 sticks of the butter to the flour mixture and toss well, using your hands or a pastry cutter, to coat the cubes in flour.
Cut the butter into the flour with your hands or a pastry cutter until the butter forms pea-sized pieces and the mixture resembles coarse meal.
Make a well in the center of the bowl, add the water, and mix until the dough comes together. It should be rough in texture but also able to hold together completely. It should not be sticky. If it's too dry or appears powdery, add more water, 1 tablespoon at a time, until the dough comes together.
Divide the dough into two pieces - 2/3 of the dough for the bottom crust and 1/3 of the dough for the top crust.
Form each piece of dough into a disk, wrap each with plastic wrap, and chill for at least 1 hour and up to 2 days.
While the crust is chilling, in a large pot, heat 2 tablespoons of the butter over medium heat.
Add the onions, the celery, and the carrots to the butter and cook until the onions are translucent, about 5-6 minutes.
Add the garlic to the veggie mixture, season with the salt and the pepper, and sauté until fragrant, about 1 minute. Transfer the veggie mixture to a plate.
Add the remaining butter to the same skillet over medium heat and melt.
Whisk the flour into the butter and cook, while stirring constantly, until the mixture is just starting to turn golden brown, about 2-3 minutes.
Gradually whisk the chicken broth into the flour mixture.
Bring the chicken broth mixture to a simmer, then add the onion mixture, the bay leaf, the rosemary, and the thyme back to the pot.
Bring the mixture to a simmer, stirring occasionally to prevent scorching, about 15 minutes.
Stir the heavy cream, the chicken, and the peas into the mixture and return to a simmer and cook, about 4-5 minutes.
Transfer the filling mixture away from the heat and allow it to cool completely.
On a lightly floured surface, roll out the 2/3 piece of dough to 1/4-inch-thickness.
Transfer the dough to the prepared springform pan and trim the edges so there is 1/2-inch of overhang all around the edge of the pan.
Chill the dough inside of the pan for at least 30 minutes and up to overnight.
Preheat the oven to 400 degrees F.
Dock the chilled dough with a fork on the base and the sides.
Line the crust with parchment paper and pie weights and bake until the crust begins to brown on the edges, about 15-17 minutes.
Carefully discard the parchment and the pie weights and return the pie to the oven until the base appears dry, about 2-3 minutes.
While the crust is warm, carefully use kitchen shears to cut away any excess crust that's still hanging over the edge of the pan.
Allow the crust to cool to room temperature.
Mound the cooled filling into the cooled crust, pressing firmly to ensure there are minimal air pockets.
Make a rounded mound on top of the filling.
On a lightly floured surface, roll out the remaining dough to 1/4-inch-thickness.
Use a rolling pin to transfer the dough to the top of the pie.
Trim away any excess dough, leaving about 1/2-inch excess all the way around the edge.
Nudge the edges of the dough down into the side of the pan so that it meets the top edge of the bottom crust.
Push the crust down a little so that the excess dough puckers outward and creates a lip.
Press the outer lip together to seal the edges, then crimp with a fork.
Brush the top of the crust evenly with the egg wash, and cut vents into the top of the pie crust.
Transfer the pie to the prepared baking sheet and bake until the crust is very golden and the filling is bubbling through the vents, about 50 minutes-1 hour. If the crust is browning too quickly or too much, lower the heat to 375 degrees F and/or tent the top of the pie with foil.
Allow the pie to cool for 20-30 minutes.
Slice and serve.