Heat the olive oil and butter in a large deep skillet or sauté pan with a tight-fitting lid over MEDIUM heat. Add the mushrooms and cook for 4 to 6 minutes, stirring occasionally, until browned and the moisture has evaporated. Add the onion, carrots, and celery and cook for 4 to 5 minutes, or just until beginning to soften.
Reduce heat to LOW and stir in the garlic, thyme, rosemary, salt, and pepper. Cook for 1 minute.
Add the rinsed rice and cook, stirring, for 1 minute.
Whisk the Dijon into the chicken broth and pour it into the skillet. Increase heat as needed to bring to a boil.
Cover, reduce the heat to LOW, and simmer for 45 minutes, or until the liquid has been absorbed and the rice is tender. Remove from the heat and let stand, covered, for 5 minutes. Uncover and stir well.
Stir in the rotisserie chicken, Parmesan, and parsley until the chicken is warmed through. If needed, place the skillet over LOW heat briefly, until the chicken is warmed through.
Taste and season with additional salt and pepper as needed. Serve with additional Parmesan cheese, if desired.