Melt 4 tablespoons of the butter in a large saucepan over medium-high heat.
Add the onion, the garlic, the celery, the carrots, the coriander, the salt, and the pepper and cook, stirring occasionally, until the onions are translucent, about 8-10 minutes.
Stir in the potatoes and the Brussels sprouts.
Cook until the vegetables are tender and beginning to brown, about 6-7 minutes.
Transfer the mixture to a bowl and reserve the saucepan.
Melt the remaining butter in the reserved saucepan over medium heat.
Add 1/2 cup of the flour and cook, whisking, until a smooth paste forms, about 30 seconds.
Reduce the heat to medium-low and whisk in the stock, the heavy cream, and the thyme.
Cook, stirring occasionally, until slightly thickened, about 8-10 minutes.
Stir in the peas and the lemon zest.
Transfer the saucepan from the heat.
Stir in the shredded chicken.
Add the cream sauce to the filling mixture and stir to combine.
Allow it to cool completely.
Preheat the oven to 400 degrees F.
Let the pie crust sit at room temperature for 10-15 minutes.
Using the extra flour, dust a work surface, and roll one dough disk into a 14-inch round.
Fit the round into a 10-inch cast-iron skillet or a deep-dish pie plate.
Trim the edges, leaving a 1 1/2-inch overhang.
Roll the second dough disk into a 12-inch round.
Spoon the filling into the bottom crust.
Place the second round of dough over the filling, trimming the edge to a 1-inch overhang.
Fold the edge of the bottom crust over the top, pressing the bottom and top crusts tightly together.
Ensure the crust rests on the edge of the pan and press the edges with your fingers to seal the loose pleats.
Cut two small vents in the top crust.
Brush the top crust with the beaten egg.
Bake until the crust is golden-brown, and the filling is bubbling, about 50-60 minutes. Shield the crust with aluminum foil after 30 minutes if it's starting to brown too quickly.
Serve hot.