In an 8-quart pot, melt butter over medium heat.
Add chopped celery, carrot, and onion. Saute until soft, about 5 minutes. Add minced (or microplaned) garlic and chopped thyme. Saute for about another 2 minutes.
Add flour and mix around. Pour in chicken stock and heavy cream. Add Tony Chachere's.
Add chicken quarters. Bring to a low boil and boil for about 20 minutes. If you can't do them all at once, you can do them in 2 batches.
When the chicken in cooked through, remove from pot and remove the chicken from the bone. You can use utensils, or let it cool for a few minutes and use your hands (I always get way more chicken when I use my hands)
Return deboned chicken to the pot. Taste broth and season as needed. Salt and pepper as desired. Reduce heat to low and let simmer for about 30 more minutes.
While the broth is simmering, make the dumplings. Combine 2 cups of bisquick and 2/3 cup of milk. At this point, the broth will be quite thin. When you cook the dumplings, it will thicken substantially.
When you are happy with the flavor of the chicken and broth, you can begin to drop the dumplings. Using a regular kitchen spoon, grab a spoonful and drop into the broth (broth should still be on low, or slightly above). Continue to drop dumplings by the spoonful until it is all used. *Note* The dumplings will puff up substantially so the dumpling dough you drop doesn't need to be a giant spoonful.
Let the dumplings simmer for 10 minutes uncovered. Resist the urge to stir them. You can LIGHTLY push them around after they have been in a few minutes, but don'd do this too much. At this point they will still be quite soft.
After 10 minutes or simmering uncovered, give them a gentle stir (don't fret if they are stuck together) and cover and let simmer for another 10 minutes.
After another 10 minutes, uncover and stir. Dumplings should be cooked throughout, light and fluffy. If not, let them simmer for another few minutes.
When the dumplings are cooked through, spoon into a bowl and enjoy!