Set the Instant Pot to sauté on medium. Add the olive oil and butter. Once butter is melted, add the onion and cook for 4 to 5 minutes, until softened and translucent. Add the garlic and cook until fragrant, about 1 minute.
Add the carrot, celery, and potato. Cook until vegetables begin to soften, about 4 to 5 minutes.
Add the cream of chicken soup, chicken broth, thyme, and bay leaf. Stir to combine.
Lay the chicken breasts on top of the mixture. Place the lid on the Instant Pot and pressure cook on high for 15 minutes. When done cooking, allow the pressure to naturally release for 10 minutes before manually releasing the remaining pressure.
Meanwhile, as the soup cooks, gently roll out the puff pastry sheet and cut it into 8 even pieces. Bake the pastry until puffed and golden, about 8 to 10 minutes. Set aside.
When the pressure is done releasing, carefully remove the lid of the Instant Pot and transfer the chicken breasts to a bowl. Shred the chicken breasts and place the shredded meat back into the Instant Pot. Add the peas and stir to combine, ensuring they heat through.
Add the heavy cream and mix thoroughly. Add salt and pepper to taste.
Serve immediately, garnishing with parsley and a square of puff pastry.