Preheat oven to 400 degrees.
Season chicken generously with salt and pepper. Sprinkle all over with Italian seasoning.
In a large cast iron skillet combine oil and butter over medium-high heat.
Once butter is melted, add the chicken and brown on each side for 1-2 minutes until you get a nice browned color. Transfer to a plate and set aside.
In the same skillet, begin preparing the sauce by adding the butter. Once melted, stir in garlic and mushrooms and sauté 2-3 minutes until mushrooms are browned and garlic is fragrant.
Stir in chicken broth, heavy cream, dijon mustard, salt, and pepper. Bring to a boil, cook for two minutes at a high boil. Remove from heat.
Return chicken to skillet. Spoon sauce over the chicken, then transfer skillet to the oven to bake for 15-20 minutes until chicken is cooked through and sauce has reduced (it should be thick enough to coat the back of a spoon).
Spoon the mushroom sauce over the chicken and serve.