In a large, deep heavy skillet over medium-high heat, warm the 1 tablespoon olive oil and 1 tablespoon (1/2 oz) unsalted butter.
Add the 1 1/2 pounds boneless, skinless chicken breasts, season lightly with kosher salt and freshly ground black pepper, and cook, stirring occasionally, until it just starts to brown on the outside, 6 to 8 minutes.
Using a slotted spoon, transfer the chicken to a plate.
Add the 2 cups peeled and finely diced carrots, 1 cup finely diced celery, and 1 cup finely diced onion to the skillet along with the 2 teaspoons fresh thyme leaves a big pinch of salt and a few grinds of black pepper. Cook over medium heat until the vegetables just start to soften, 4 to 7 minutes.
Toss in the 8 ounces white or cremini mushrooms and cook until they start to brown and release their juices, 4 to 6 minutes.
Sprinkle the 1/4 cup all-purpose flour over the vegetables and stir until completely coated.
Pour in 1 cup of chicken stock and cook until the liquid starts to thicken, about 15 seconds.
Return the chicken back to the skillet, then pour in the rest of the stock and increase the heat to high. Bring the mixture to a boil.
Once the mixture is bubbling, reduce the heat to medium-low and simmer until the liquid has reduced and thickened, 10 to 15 minutes. It should look like a slightly watery stew at this point.
Reduce the heat to low and stir in the 1/4 cup heavy cream. Cook over low heat until the sauce reaches the desired consistency.
Season to taste with salt and pepper and sprinkle with fresh thyme or fresh parsley to serve.