Prepare the ‘bouquet garni’ by assembling the parsley, thyme, and rosemary sprigs on a clean work surface and securing with kitchen twine, as shown. Securely tuck the bay leaf inside the twine to hold. Set aside.
Press the “Sauté” button on the Instant Pot® and adjust to the highest setting. Add the olive oil to the pot, along with the garlic, onion, carrots, and celery. Season with salt and black pepper, to taste, and stir to combine. Cook, stirring occasionally, until the vegetables soften and develop a bit of color, approximately 4-5 minutes. Turn the unit off.
Add the chicken broth to the large pot, along with the chicken thighs and breasts. Add the raw chicken down into the liquid with a wooden spoon or spatula until fully submerged.
Add the ‘bouquet garni’ on top and gently push into the liquid, as well. Add the lid and lock into place. Set the pressure release valve to “Sealing” and select the “Manual” button on the high setting. Adjust cook time to 10 minutes.
When the cook time is complete, allow the pressure to release naturally for 5 minutes before doing a quick release of any remaining steam. Turn the unit off.
Carefully remove the lid and transfer the chicken pieces to a large, rimmed plate. Shred the meat using two forks and set aside.