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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.
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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.
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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.
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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.
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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.
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6
Pour filling into the muffin cups.
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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.
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8
Bake in the preheated oven until golden brown, 15 to 25 minutes. Let sit for at least 5 minutes before serving.
