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.
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.
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.
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.
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.
Pour filling into the muffin cups.
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.
Bake in the preheated oven until golden brown, 15 to 25 minutes. Let sit for at least 5 minutes before serving.