Before you make the crust, cut the butter for the crust into small cubes and set on a plate. Place in the freezer for 15-30 minutes so it gets VERY COLD. While you do this, you can get all of your other ingredients out and ready to go.
To make the crust: Add the flour and salt to a food processor and pulse a few times to combine. Add the cold butter cubes and pulse again for 20-30 seconds until pie dough starts to resemble tiny peas. Add the egg yolk, then add in 2-3 tablespoons of cold milk and herbs (if using) and pulse until the dough comes together, resembling small beads. The dough should still be somewhat crumbly; I recommend squeezing a small amount of dough between your fingers; it should stick together well, but if it doesn’t and is very crumbly, add a little more milk, ½ tablespoon at a time, until it just comes together. If your dough is too wet, add ½ tablespoon flour until it comes together. I usually don’t mess with ratios too much because this is what works for me. Do not overwork the dough.
Place dough onto a well-floured surface and form into a disk shape, then wrap with plastic wrap or in a reusable silicone bag. Chill in the fridge for at least 30 minutes to 1 hour or up to 2 days. Dough can also be frozen for up to 2 to 3 months if well wrapped in both plastic and then foil.
Cook the chicken: In a large 10-inch deep oven-safe skillet, warm olive oil over medium-high heat. Once oil is hot, add in the chicken and season with salt and pepper. Cook until chicken is no longer pink, about 5 minutes, then transfer chicken to a medium bowl or large plate and set aside. Don’t worry if this isn’t completely cooked, as it will finish in the oven. See notes for shortcuts.
Cook the veggies: Reduce heat to medium, then add the butter to the same skillet. Once butter is melted add in the onion, celery, carrots and saute, stirring frequently, until veggies have softened, 5 minutes. Stir in the garlic and thyme and cook until fragrant, 1 minute. Next whisk in the flour to coat the veggies, then slowly add in the heavy cream, broth, salt and pepper, whisking until no flour lumps remain. Stir in the cooked chicken and peas and cook the mixture, stirring occasionally over medium-low heat, until thick, 10 minutes. Taste and adjust seasonings as desired, such as adding more thyme, salt or pepper. Set mixture aside to cool while the oven preheats.
Preheat the oven to 425 degrees F.
Roll out pie dough: Place pie dough on a VERY well-floured clean surface and use a well floured-rolling pin to roll the dough into a round shape about 10 to 11 inches in diameter. Fold dough in half to help transfer over the cooled chicken pot pie skillet then lay over the top of the filling. Working quickly, use a fork to crimp the edges. Lightly brush the pie dough with the heavy cream. Use a small sharp knife to make a slit or two in the middle pie crust to allow steam to escape.
Transfer the skillet to a large baking sheet, then place in the oven and bake for 30 to 40 minutes until the crust is golden brown and filling is bubbly. Top crust with flaky sea salt and fresh thyme leaves.