Press Sauté on the Instant Pot and heat the olive oil and butter until the butter melts and foams. Add the onion, carrots, and celery, and cook for 4 to 5 minutes until the onion turns soft and translucent, then add the garlic and cook for 30 seconds until fragrant.
Press Cancel to turn off the Sauté function. Pour in about 1 cup (240 ml) of the chicken broth and scrape the bottom of the pot with the wooden spoon to lift any browned bits, making sure nothing is stuck to the base.
Pour in the remaining chicken broth and the water. Add the thyme, crushed rosemary, bay leaves, salt, and black pepper, and stir once to combine. Nestle the chicken breasts into the liquid so they are fully submerged.
Close the lid and turn the valve to the Sealing position. Select Pressure Cook or Manual on high pressure and set the time to 8 minutes. The pot will take roughly 12 to 15 minutes to come up to pressure before the countdown begins.
When the cook time ends, let the pressure release naturally for 10 minutes. After 10 minutes, carefully move the valve to Venting to release any remaining pressure, keeping your hand and face away from the steam, then open the lid once the pin drops.
Lift the chicken breasts out of the pot and set them on the cutting board. Remove and discard the bay leaves, then shred the chicken with two forks and set it aside while you cook the noodles.
Press Sauté again and bring the broth to a gentle boil. Add the egg noodles and cook uncovered for 6 to 8 minutes, stirring occasionally, until they are tender but still have a little bite. Taste the broth and add more salt if it needs it.
Stir the shredded chicken back into the pot and let it warm through for a minute. Turn the pot off, then stir in the lemon juice if you are using it. Ladle into bowls and top each one with chopped fresh parsley.