In the bowl of a food processor with the regular blade attached, add 1 cup of the flour and 1/4 teaspoon of the kosher salt and pulse lightly to combine.
Add 1/3 cup of the cold diced butter and pulse until it is lightly mixed and there are pea-sized balls of butter.
Add the egg yolk and 1 tablespoon of the milk and process again until the dough is crumbly and somewhat wet. Add the extra milk, a little at a time, until the desired consistency is reached. Be careful not to let the dough overmix; the dough should not become a ball.
Using the extra flour, flour a work surface very well.
Pour the dough onto the prepared surface and gather it into a disk.
Wrap the dough disk in plastic wrap and chill it in the refrigerator for at least 1 hour.
In a large pot over medium-high heat, add the oil.
Add the chicken, the remaining kosher salt, and the pepper to the oil and cook, while stirring, until the chicken cubes are cooked through, reaching an internal temperature of 165 degrees F, about 6-8 minutes.
Transfer the chicken cubes to a plate.
In the same pot over medium heat, melt the remaining butter.
Add a portion of the remaining 1/4 cup flour and whisk it into the butter.
Add a portion of the remaining 1 3/4 cups milk and whisk to combine.
Continue to add more of the flour and the milk to the melted butter mixture a little at a time, alternating between the 2, whisking vigorously to remove any lumps.
Bring the sauce mixture to a slight boil.
Reduce the heat to low and cook, while stirring occasionally, until the sauce thickens slightly, about 3-4 minutes.
Add the garlic powder, the onion powder, more of the remaining kosher salt, and a generous amount of the pepper and stir to combine.
Add the cheddar cheese and stir until completely melted.
Add the broccoli, the carrots, and the cooked chicken and stir to combine and completely coat.
Preheat the oven to 375 degrees F.
Coat a 10-inch-deep skillet or a large, deep pie pan with cooking spray.
Add the filling mixture to the skillet or pan and smooth out the top.
Using the extra flour, dust a work surface very well.
Unwrap the pie dough and place it on the prepared work surface.
Roll out the pie dough so it will completely cover the pie pan.
Place the dough on top of the filling, pressing the edges to seal.
Cut a few slits in the top of the crust to allow steam to escape.
With the remaining beaten egg, brush the top of the crust all over.
Bake until the crust is golden and the filling is bubbly, about 45-55 minutes.
Sprinkle the crust with the coarse sea salt.
Serve.