Preheat the oven to 375 degrees.
Heat up the olive oil in a large skillet over medium-low heat.
Add the onion, minced garlic, frozen pas, mushrooms, and carrot pieces into the skillet. Stir and cook for about 3 minutes.
Add in the cooked chicken, cream of chicken, and milk. Stir to combine.
Turn the heat down low and let the chicken mixture cook until it starts bubbling. Immediately remove from the heat.
Cut the puff pastry up into 8 even pieces.
Coat the inside of four ramekins by sliding the butter on the inside of each.
Place a piece of puff pastry inside a ramekin, pressing it in so it covers the entire inside (like a mini pie). Repeat for the other ramekins.
Scoop the chicken mixture evenly into the four ramekins.
Lightly stretch a piece of puff pastry and cover the top of the ramekin, tucking and pressing the edges of the pastry dough over the sides so it is completely covered. Repeat for all of the ramekins.
Make small slits with a serrated knife in the top of your mini chicken pot pies.
If desired, whisk the egg in a small bowl. Using a pastry brush, spread the egg on top.
Place all of the ramekins on a sheet pan, then put the sheet pan in the oven.
Bake for 30 minutes.
Serve with a side of roasted vegetables!