Preheat oven to 375 degrees F. In a 10-inch cast iron skillet, add oil and sauté the diced chicken with salt and 1/4 teaspoon black pepper, and garlic until chicken is no longer pink, about 6-8 minutes.
Remove from skillet and put in a bowl. There should be some fat and juices leftover in the pan from the chicken; add diced onion to the skillet with a big pinch of salt. (If you are using pre-cooked chicken, add 1 tablespoon oil). Cook over medium heat, letting sit for 5 minutes without stirring so you get some browning.
Then add 2 tablespoons white wine, 2 sprigs of thyme and stir, deglazing the pan and continuing to cook for 8-10 minutes, until onions are very soft. Remove from heat, discard the thyme sprigs.
Add chicken back to skillet, then stir in remaining 2 tablespoons of white wine, chicken stock, sour cream, cream cheese, 1 cup cheese, black pepper and minced thyme from the remaining sprig. Spread to make smooth, then top evenly with remaining 1 cup of cheese.
Bake for 15-18 minutes, until cheese is browned and bubbly. Sprinkle with extra thyme and a few cracks of black pepper. Serve hot.