Preheat the oven to 375°F.

For the crust: In the work bowl of a food processor, combine the flour, cheese, butter, salt, and a few grinds of pepper; pulse until crumbly. Slowly pulse in 4 tablespoons of ice water; add up to 2 more tablespoons of ice water until the dough comes together and is the texture of wet sand. Transfer the dough to a piece of plastic wrap and wrap tightly. Refrigerate until ready to use, at least 30 minutes.

Meanwhile, for the filling: Melt the butter in a large pot over medium heat. Add the onion, carrot and broccoli and cook, stirring, until the vegetables soften, 3 to 4 minutes. Sprinkle the flour over the vegetables and stir, then stir in the broth. Increase the heat to medium-high and bring to a simmer. Whisk until no lumps of flour remain, then continue cooking until thickened and slightly reduced, 4 to 6 minutes. Stir in the chicken, thyme, heavy cream, and a pinch each of salt and pepper. Transfer the filling to a 9-inch pie dish and set aside to cool.

Whisk the egg with 2 tablespoons water in a small bowl until smooth.

Transfer the dough to a lightly floured surface. Using a lightly floured rolling pin, gently roll out the dough into a 12-inch round. Center the dough round over the filling in the pie dish. Trim any overhanging dough, then press the edges against the pie dish to seal, crimping as desired. Brush the crust with the egg wash. Use a paring knife to make a few small steam vents in the crust.

Bake the pot pie until the crust is golden, 24 to 30 minutes. For more browning, broil for an additional 1 to 2 minutes. Let cool for 5 minutes before serving.