Melt the butter in a large Dutch oven over medium heat. Add the sliced onions, carrots, and celery. Sauté until softened, about 5 minutes. Then add the garlic and continue to sauté for an additional minute, or until fragrant.
Add the tomato paste and flour to the pot. Stir to combine and continue to cook for about 2 minutes.
While continually stirring, add the chicken broth, thyme, dry English mustard, bay leaf, and chicken breast chunks. Bring to a gentle boil and simmer until the chicken is fully cooked, about 10 - 12 minutes. Use an instant thermometer to test and make sure the chicken has reached 165F.
Add the half-and-half and bring back to a low simmer. Next add the spinach, gnocchi, and simmer the amount of time that is listed on the packaged directions.
Taste, and season with salt + pepper to your preference. Serve immediately with cheddar, chives, and parmesan as a garnish.