Begin by preheating your oven to 375 degrees.

For the filling, combine 4 tablespoons of softened butter with flour in a small bowl.

Set this mixture aside.

In another bowl, dissolve cornstarch in milk and set it aside.

In a large saucepan, melt 2 tablespoons of butter and sauté the onion, carrot, and celery over medium-high heat for approximately 3 minutes.

Add the broth and potatoes, bringing it to a boil. Cover and simmer for about 10 minutes, or until the vegetables are just tender.

Incorporate the butter and flour mixture, stirring constantly as it comes to a gentle boil.

Pour in the milk and cornstarch mixture, bringing it to a boil. Season with pepper to your taste. Stir in the chicken and peas, then cover and let it cool.

For the crust, Place the butter, flour, and salt in a food processor and pulse until the mixture looks like coarse sand, with some small butter pieces no larger than peas.

This should take about 15 to 20 pulses, each lasting one second.

Add ice water one tablespoon at a time through the food processor's feed tube, pulsing 4 to 5 times after each addition. Stop adding water once the mixture begins to form large clumps and pulls away from the sides of the bowl.

Transfer the mixture to a work surface and use your hands to bring it together into a single lump. Divide the dough in half, quickly shape each half into a ball, and flatten them into 4-inch disks. Wrap each disk in plastic wrap or place them in plastic bags and refrigerate for at least an hour, or up to four days. Remove the dough from the fridge 3 to 4 minutes before rolling so it can soften slightly.

Lightly flour your work surface and rolling pin. Roll out one disk of dough from the center outward, turning it a quarter turn after each roll, until you have a circle about 12 inches in diameter. If the edges start to split, pinch them together and continue rolling. Use a large round cookie cutter slightly bigger than your ramekins to cut out the crusts, and let them rest for 5 minutes.

Place a small amount of dough at the bottom of each ramekin, just enough to create an extra crust layer.

After shaping the dough in each ramekin, fill them with the pie filling. Add the top crust and pinch it onto the ramekin. Make slits on top for ventilation and bake for 30 minutes. After 30 minutes, lightly brush with egg and bake for an additional 5 minutes. Remove from the oven and allow it to rest for at least 15 minutes before serving!