In the pressure cooker, place the chicken breasts inside with a ½ cup of water and steam cook on high for 15 minutes. When finished, rapid release the steam, take the chicken out and shred. Dispose of the leftover chicken water in the pressure cooker.
While the chicken is cooking, chop the shallots, bell pepper, green beans, and rosemary. Also drain and rinse the cannellini beans.
When the pressure cooker is empty from the chicken, place the cooker on the saute setting, drizzle 1 tbsp of olive oil on the bottom and add in the shallots, bell pepper, and green beans into the cooker. Saute for 5 minutes or so, until the shallots and bell peppers begin to slightly brown.
Turn the cooker off of the saute setting, and add in the shredded chicken, chicken stock, cannellini beans, rosemary, and red pepper flakes. Pressure cook on high for 10 minutes.
While the pressure cooker cooks, in a pasta pot, boil water for the noodles. Cook according to the instructions on the pasta box for al dente. Cooking the pasta before adding to the soup will keep the boodles from becoming soggy as well as from soaking up the chicken stock from the soup.
When the pressure cooker finishes, rapid release the steam and add in the whole milk, mascarpone cheese, and parmesan cheese. Stir until completely combined. Add in the noodles. Serve immediately, top with shredded mozzarella cheese if desired.