To make the dough: 

1. In a food processor, combine the flour, sugar, and salt. Add the cold butter and pulse just until combined. While the processor is running, slowly stream in the cold water and blend until the dough becomes crumbly.

2. Turn the dough out onto a clean work surface and shape it into a disc. Wrap the dough in plastic wrap and refrigerate for at least 2 hours or overnight.

To make the filling: 

1. In a large, shallow pan, heat 2 tablespoons of the butter over medium-high heat. Add the onion, carrots, and celery and sauté, stirring, until tender, 4–5 minutes.

2. Add the mushrooms and garlic to the pan and cook, stirring constantly to prevent the garlic from burning, for about 1 minute. Add the wine and cook until the pan appears dry, about 1 minute more.

3. Add the remaining butter to the pan and cook until completely melted. Add the flour and mix well to combine with the veggies (it should begin to thicken). Cook until the flour fully combines with the butter and the mixture is thickened, about 1 minute more (we want to ensure that all of the raw flour taste has cooked out).

4. Add the chicken stock to the pan and stir until everything is combined. Let simmer until the mixture starts to thicken, 2–3 minutes. Take your time with this step—this is where the filling gets its texture! Add the peas and the chicken to the filling and stir to combine, then stir in the bay leaf and fresh thyme sprigs.

To assemble and cook the pot pie: 

1. Preheat the oven to 400ºF.

2. Roll the pie crust out to ¼-inch thickness.

3. Remove and discard the herbs from the filling, then season with salt and pepper to taste. If the filling isn’t already in an oven-safe vessel, transfer it to one now.

4. Top the filling with the pie crust and brush it with the beaten egg. Bake until golden brown, about 20 minutes. Let cool for 5 minutes before serving.