Set your Instant Pot (with at least a 6 quart capacity) to the high sauté function. Add butter and melt. Add carrots, onions, and celery and sauté (stirring often) for 2 minutes. Add garlic and cook until fragrant (while stirring), about 30 seconds.
Add broth, water, rosemary, dill, salt, pepper, and bay leaves. Stir to combine.
Add chicken and turn off Instant Pot.
Place the lid on and lock. Steam release knob should be set on "sealing". Cook on manual setting (high pressure) for 12 minutes. When finished cooking, quick release the pressure.
Once the pressure has completely released, remove the lid and carefully remove chicken with tongs and transfer to a cutting board.
Turn off Instant Pot and set to the high sauté function. Add noodles and stir to combine. Cook noodles until tender (stirring occasionally), about 6 minutes.
While noodles are cooking, remove skin from chicken and discard. Remove chicken from bones and shred. Discard bones.
Add shredded chicken to the noodles and stir to combine. Remove and discard bay leaves before serving. Season with additional salt and pepper, if desired.