Preheat oven to 425°.
Heat a large pot on medium heat and add olive oil and butter. Stir in onions, carrots, celery, salt, and pepper. Toss to coat, and allow to cook for 5 minutes.
Stir in garlic and cook for 30 seconds; stir in flour to coat the vegetables. Cook 3–5 minutes, stirring to create a roux.
Using a biscuit or cookie cutter, cut 12 "biscuits" out of the thawed puff pastry sheet. Place them on a parchment-lined baking sheet, brushing tops with beaten egg. Bake 10–12 minutes or until golden brown.
Add potatoes, stock, and half-and-half to the soup, stirring and allowing the mixture to come to a boil. Reduce heat to medium low, add chicken, peas, and corn, and cover, simmering 10–15 minutes.
Top each bowl of soup with puff pastry.