Preheat the oven to 400°F (200°C).
In a skillet, sauté the diced onions until translucent.
Sprinkle flour over the onions and stir for 1-2 minutes to cook off the raw taste.
Gradually whisk in the chicken broth and milk, stirring until smooth.
Add the diced chicken, peas, carrots, thyme, salt, and pepper to the mixture.
Remove from heat and allow the filling to cool slightly.
Roll out the pie crust and cut into rounds for the mini pies.
Place the dough rounds into the greased muffin tin or pie pans, pressing gently to fit.
Spoon the chicken filling into each pie crust, being careful not to overfill.
Cut additional rounds of dough for the tops and place over the filling, sealing edges.
Brush the tops with beaten egg for a rich, golden-brown finish.
Bake in the preheated oven for 20-25 minutes, until golden brown.
Allow the mini chicken pot pies to cool slightly before serving.