Preheat your oven to 425 degrees F and line a baking sheet with foil.
Heat the butter (1 tablespoon) with the olive oil (1 tablespoon) in a large skillet over medium-high heat. Season both sides of the chicken thighs with kosher salt and black pepper, liberally. Sear the chicken skin-side-down for 3-4 minutes, or until it's browned and releases easily from the pan. Remove the chicken to the prepared baking sheet (skin side up). Bake for about 30 minutes, or until internal temperature of the chicken is 165 degrees F.
While the chicken is baking, make the mushroom sauce. Add the sliced onion and mushrooms (16 oz.) to skillet along with a pinch of salt and pepper and the fresh thyme (1 tablespoon), and sauté over medium-high heat, stirring only occasionally, until mushrooms are browned (5-10 minutes).
Add the flour (2 tablespoons) to the mushroom and onion mixture and stir to coat, continuing to cook for another one minute.
Gradually pour in the chicken broth (1 cup). Stir and continue to cook over medium heat, until sauce has thickened. Stir in the heavy cream (1/2 cup). Turn off heat and taste and adjust seasoning if necessary.
When the chicken is done cooking, remove from oven and place in the skillet with the mushroom sauce, along with a spoonful or two of the juices from baking.
Serve chicken with mushroom sauce on top.
I like serving this meal with cooked rice, mashed potatoes, pasta, crusty bread, or something else that's great for mopping up all that delicious sauce.
Lighten it up: use whole milk or half and half instead of heavy cream.
To use boneless, skinless chicken thighs, I recommend cooking them on both sides in the skillet until they're completely done (about 6-8 minutes on each side), then removing to a plate, making the sauce in the skillet, and transferring them (and their juices) back into the skillet. If you want to use boneless, skinless chicken breasts, I recommend using chicken breast cutlets (which are thinner) and using the same method as boneless skinless thighs (be careful not to overcook them!).
You can omit the flour to make this lower-carb and grain-free. If you do this, the sauce won't be as thick. You can stir in 1/2 cup-1 cup of grated parmesan cheese when you add the cream until it melts and thickens the sauce (if you do this, go easy on the added salt since the cheese is salty). If the sauce still seems too thin, keep simmering it until it's reduced a bit.