For the pie dough:
Place the flour in the bowl of a food processor fitted with a metal blade, along with the butter and a large pinch of salt. Pulse a few times until the butter is broken up and the mixture has a mealy texture. While pulsing, add the water one tablespoon at a time. Pulse until the dough starts to come together into a ball. Turn the dough onto a floured surface, fold over a couple of times to knead and shape into a disk. Wrap the dough in plastic and refrigerate for at least an hour.
For the pot pies:
Preheat oven to 350°F.
Melt 1 tablespoon of the butter in a Dutch Oven set over medium heat. Add the onion, carrots, and celery and sauté until the onions are translucent and the vegetables are just tender, about 6-8 minutes. Season with a pinch of salt and pepper. Add the garlic and cook until fragrant, about 1 minute longer.
Add the remaining tablespoon of butter and stir until melted. Sprinkle the flour evenly over the surface of the pot, then stir until completely combined. Cook, stirring constantly, for 2 minutes. Add the broth to the pot, and bring to a simmer, stirring frequently. Season again with salt and pepper. Add the bay leaf. Simmer, stirring frequently, until the mixture thickens enough to coat a spoon or spatula, about 5 minutes. Add the cream and bring to a simmer. Add the shredded chicken and peas, and cook for 5 minutes longer. Remove the bay leaf and let the mixture cool.
Once the pot pie mixture is cool, divide the filling between four 8 oz. Mini Cocottes or ramekins.
Remove dough from the refrigerator and place on a floured work surface. Roll out dough until it is about 6 x 12 inches long. Using a pastry roller or a pizza cutter, cut the dough into 4-inch long strips that are about 1/2 inch wide. You will need about 24 strips. Arrange 3 strips across the top of each cocotte. Form lattice by arranging 3 more strips diagonally across the first strips. Use the excess dough to form a border around the edges of the cocotte, attaching the loose ends of the lattice. In a small bowl, whisk the egg and water to combine. Brush the egg mixture over the surface of the dough.
Place the cocottes on a lined sheet pan and place in preheated oven. Bake until the filling is bubbly, and the crust is golden brown, about 15-20 minutes. Remove from the oven and let rest 5 minutes before serving.
