If cooking chicken, pat chicken dry with paper towels and season with the 1 teaspoon salt and ½ teaspoon pepper on both sides. Add more if you like.
Heat oil in 5 qt (or larger) soup pot over medium-high heat, then place 3-4 chicken thighs into the pot to sear, about three minutes on both sides. Make sure the thighs aren't over crowding the pot. Remove thighs and place on a plate. Repeat with the second batch.
Lower heat to medium and add onion, carrot, and celery along with a pinch of salt and pepper. Stir occasionally and cook for 4 minutes to soften the veggies. Add garlic and thyme and stir together until garlic is fragrant, about 30-45 seconds.
Toss butter into the pot and allow to melt, stirring all the while. Shake in the flour and stir to coat the vegetables.
Splash in chicken broth ½ cup at a time, stirring well between each addition. After 1 ½ cups, pour in the rest and stir the roux and broth together. Add water, then add chicken back to the pot. Bring heat to medium high and allow to come to a boil for 3 minutes, then turn down heat to medium-low to simmer covered for 40 minutes.
Make your dumplings just before the 40 minutes end; in a medium bowl mix flour, baking powder and salt together. Add in buttermilk and herbs and stir until flour is just incorporated. We want a slightly to somewhat wet looking dough, so splash in up to ¼ cup more buttermilk if needed.
Before dropping biscuits in, pull out the chicken if you want to shred them with a fork. Disregard if you already have pre-cooked, shredded chicken. Add chicken back in, along with heavy cream, and stir well.
Use a spoon to drop spoonfuls of dumpling batter into the pot. You can move them around and mess with them while in the pot, this results in a thicker stew and smaller sized dumplings. Cook covered for 5 minutes then flip them over using a large spoon and cook uncovered for 15 minutes.
Taste and adjust seasoning before serving. Serve hot.