Set the Instant Pot to the sauté setting and turn to high. Add the olive oil, carrots, garlic, onion and celery. Cook until tender, about 3-4 minutes.
Stir in the dried rosemary and Italian seasoning and cook for one additional minute.
Wash chicken and pat dry. Remove skins from chicken. Season the chicken well with salt and pepper and add to the pot.
Add broth, water, bay leaves, and ½ teaspoon of both salt and pepper to the pot.
Select the manual setting on your Instant Pot and set the time for 10 minutes on high pressure.
While the soup is cooking, make your noodles. Combine the flour and salt in a bowl, then, make a small well in the center. Crack the eggs into the “well”, and beat the eggs, gradually brining in some of the flour mixture until well combined and a solid ball of dough is formed.
Working with your hands turn the dough out onto a well-floured surface and knead the dough, adding more flour as needed to prevent it from sticking to the surface and/or your hands. (Note – try not to incorporate more flour than you actually need. Too much flour can lead to dry noodles.) Knead the dough until you have a smooth ball (around 5-10 minutes).
Wrap the dough with plastic wrap and chill it for 15-30 minutes (or up to overnight if making noodles ahead of time).
Remove the chilled dough from the fridge, and divide it into two. Using a rolling pin, roll out each ball individually as thinly as you can (the noodles will plump up a bit when cooking) and then cut with a pizza cutter or very sharp knife. Try to cut them evenly to ensure that they all cook evenly.
Place cut noodles on a pan or drying rack until ready to cook.
Once your soup is finished cooking, quick-release the pressure (according to your manufacturer’s directions).
Remove the chicken and remove the meat from the bone (it should come off pretty easily). Shred the chicken and set it aside.
Return the setting on your Instant Pot to the high sauté setting. Bring the soup to a boil and add the pasts. Cook, uncovered until pasta becomes tender.
Stir in chicken, lemon and a big handful of basil. Season with salt and pepper to taste.
Serve immediately with crusty bread.