Step 1
Place chicken breasts into a large pot and cover with chicken broth; bring to a boil. Reduce heat to medium-low and simmer until no longer pink in the center and the juices run clear 20 to 25 minutes. Take chicken out of broth; pull apart or chop as desired. Keep broth hot.
Step 2
Melt 3 tablespoons butter and olive oil in a large Dutch oven or stockpot over medium-high heat. Add potatoes carrots corn celery peas and garlic. Saute until onions turn soft and translucent about 15 minutes. Add 3 teaspoons salt 2 teaspoons black pepper rosemary nutmeg thyme and sage. Cook and stir until combined into the vegetable mixture about 1 minute.
Step 3
Melt remaining 2 tablespoons butter in the vegetable mixture. Sprinkle flour on top. Cook stirring constantly until vegetables are coated about 1 minute. Ladle in the hot chicken broth 1 cup at a time stirring constantly until pot is sufficiently filled; you may not need all of the broth.
Step 4
Stir heavy cream into the pot; turn up heat slightly. Add the pulled chicken. Season with salt and pepper. Cook stirring occasionally and scraping the bottom until flavors blend about 5 minutes. Reduce heat and simmer uncovered until filling has thickened about 20 minutes. Let cool for about 5 minutes.
Step 5
Preheat the oven to 375 degrees F (190 degrees C). Grease two 12-cup muffin tins. Line 2 baking sheets with aluminum foil and set the muffin tins on top.
Step 6
Pour filling into the muffin cups.
Step 7
Roll puff pastry onto a work surface and smooth it out a bit. Cut into 24 pieces slightly larger than the muffin cups. Place pastry pieces on top of the filling in the muffin cups letting about 1/4 inch of the pastry hang off the edges. Brush tops of the pot pies with the whipped egg. Season with salt and pepper. Cut some slits for venting.
Step 8
Bake in the preheated oven until golden brown 15 to 25 minutes. Let sit for at least 5 minutes before serving.