Using the Saute setting on instant pot, brown chicken in oil on both sides. Transfer to a plate. Saute carrots, celery, onion, and garlic in remaining drippings until slightly softened, about 5 minutes.
Add chicken back to instant pot along with rosemary, and chicken broth. In a small bowl, whisk half-and-half with flour and add to pot. Stir to combine. Cook soup on High Pressure setting 20 minutes. Then turn heat off and let steam release naturally for 15 minutes.
Remove chicken from instant pot and place on a cutting board. Set instant pot to Saute mode and bring to a boil. Add noodles to soup and let cook until tender, stirring occasionally. Meanwhile, use a fork to shred the cooked chicken.
Add chicken back to pot along with lemon juice and season with salt and pepper to taste. Serve soup hot and enjoy!