To begin, salt and pepper the diced with a half teaspoon each. Heat the olive oil and a tablespoon of the butter in a Dutch oven over medium heat. Add in the chicken and cook until opaque and lightly golden in some spots.
Scoop the chicken out to a dish and set aside for now.
Add in the remaining butter along with the carrot, celery, onion, and garlic. Cook over medium heat until the onion and garlic are tender.
Stir in the flour and cook for one minute.
Add in the chicken broth and half & half along with the remaining half teaspoon each of salt & pepper. Stir together and bring to a gentle boil.
Add in all of the remaining ingredients and finally add the chicken back into the pot. Let everything simmer for about 15-20 minutes, until the gnocchi are cooked through.