Make the stuffing: Set a rack in the center of the oven and heat to 400°F. Grease a 9-inch deep-dish pie pan and a 12-by-12-inch piece of parchment paper with cooking spray and set aside with another, ungreased 9-inch deep-dish pie pan.
Line a half sheet pan with parchment paper. Place bread cubes onto prepared tray and bake until dried out, about 8 minutes. Remove from oven and transfer to a large bowl.
Heat an 11-inch skillet over medium heat for about 3 minutes. Add 2 tablespoons of the butter and cook until foamy. Add 3/4 cup of the diced onion, 1 cup of the diced celery, the rubbed sage, dried thyme, 1 teaspoon of the salt, and 1/2 teaspoon of the pepper. Cook, stirring occasionally, until soft and fragrant, about 5 minutes.
Add 1/4 cup of the chicken stock to deglaze the pan and let cook for 30 seconds. Add to the bowl with the bread cubes. Add an additional cup of the chicken stock to the bowl and stir well so that the bread cubes absorb all the liquid. Add the egg to the mix and stir to combine.
Pour all of the stuffing into the greased pie pan and, using the back of a spoon or lightly dampened (and clean) hands, press the mixture down to equally distribute on the bottom and sides of the pan. Place the parchment paper, sprayed side-down, on top of the stuffing, then place the second pie pan on top of the parchment, and press down firmly until the stuffing reaches the edges of the pie pan.
Place the stacked pie pans on the middle rack of the oven and bake for 30 minutes. After 30 minutes, remove the top pie plate and parchment paper, and bake until golden and crisp on the edges, about 10 minutes more.
Make the filling: Meanwhile, put 1 tablespoon of the oil into your now-cleaned 11-inch skillet and place over medium-high heat until it shimmers. Add the turkey, season with 2 teaspoons of the salt, and cook, turning occasionally, until the turkey starts to brown, is no longer pink on the middle, and brown bits have started to stick to the bottom of your pan, about 5 minutes. Remove the turkey from the skillet and set aside in a medium bowl.
Lower the heat to medium and add the remaining 1 tablespoon of oil to the skillet; heat until it shimmers. Add the remaining 3/4 cup onion, the remaining 1 cup of celery, the mushrooms, carrots, garlic, 1 teaspoon salt, 1 teaspoon pepper, fresh sage and thyme, and stir to combine. Cook, stirring occasionally, until the vegetables have started to soften, 5 to 6 minutes. Add 2 tablespoons of the butter and cook until it has melted. Stir in the flour and cook for 1 minute. Add the wine to deglaze the pan, scraping the bottom of the pan until it is almost dry, about 1 minute. Gradually stir in the remaining 1 1/2 cups of stock and the milk. Bring to a simmer, stirring constantly, and cook until the sauce thickens, about 2 minutes. Remove from the heat and stir in the peas and turkey. Set aside.
Make the potatoes: In a medium saucepan, put the potatoes, 2 teaspoons of salt, and enough cold water to cover. Set over high heat, cover, and bring to a boil. Uncover, drop the heat to maintain a simmer, and cook until tender, 10 to 15 minutes.
Meanwhile, combine the half and half and remaining 4 tablespoons butter in a microwave-safe container and nuke until warmed through and the butter has melted, about 35 seconds. Melt the remaining 1 tablespoon butter and set aside.
Drain the potatoes and return them to the saucepan. Mix the potatoes using a hand mixer until they are just mashed, about 30 seconds, then add the hot half and half mixture, the remaining 1/2 teaspoon salt, and 1/4 teaspoon pepper. Mix just until creamy and combined, being careful not to over mix, about 30 seconds. Gently fold in the egg yolk.
Assemble the pie: Pour the turkey filling into the stuffing crust, top with the potatoes, and gently spread to cover the filling using a rubber spatula. Brush the potatoes with the melted butter.
Bake on the center rack in the oven until the filling is bubbly and the potatoes have started to brown on top, about 25 minutes. Let rest 15 minutes before slicing and enjoying.