Heat the olive oil and butter in a large skillet over medium high heat. Add the onion, thyme and a pinch of salt and pepper. Sauté a few minutes to soften.
Add the broccoli and sauté until it begins to turn bright green, keeping it moving so it does not burn. When the broccoli is close to being crisp tender add the mushrooms and cook until they are tender and hot.
Garnish with grated parmesan cheese and serve immediately.