Arrange the chicken in a single layer on the bottom of a pot. Scatter salt over the top.
Pour in enough cool water to cover the chicken by an inch or so.
Place the pot of chicken on the stovetop over medium-high heat. Bring the water to a boil. As soon as the water comes to a boil, reduce the heat to low, cover the pot, and let the chicken simmer. Remove any foam with a spoon that accumulates at the top during cooking.
Check the chicken after 10 minutes. It is done when no longer pink and an instant read thermometer inserted in the thickest part of the meat registers 170 degrees F. (I find the chicken typically cooks in 10-12 minutes).
Remove the chicken from the liquid and place on a cutting board to cool. When cooled, cut the chicken into bite-sized pieces.
Heat a medium sized pot on medium-high heat. Add the olive oil.
Add the chopped onions, carrot and celery to the pot and cook for 8 minutes, stirring occasionally.
Add the garlic and cook for a minute more.
Pour in the chicken broth and bring to a boil. Reduce the heat to low, add in the chicken pieces and simmer for 10 minutes.
Add the pasta to the pot, stir and cook until pasta is tender, about 5 minutes.
Add salt and pepper to taste.