Preheat oven to 350. In a 12 inch skillet, melt butter and vigorously stir in flour. Add in celery and carrots, and stir frequently for about 4 minutes total.
Pour and stir in broth until broth is fully mixed into the mixture and you don't see any bits of flour. Add in chicken, thyme, parsley salt and pepper. Simmer for about 5 minutes on low
While the chicken is simmering, mix in all the dumpling ingredients minus the cheese. combine with a fork until you have a ball of dough. It should be just wet enough to stick together. If its too doughy add in more flour. It ifs not sticking add in milk 1 tbsp at a time.
Spoon in dumplings into the skillet and place in the oven. Cover dumplings with cheese and top entire skillet with a little more salt, pepper, thyme and parsley. Cook for about 25-30 minutes until the chicken internal temperature is 165 F. Let cool for about 5 minutes and serve.