Heat the olive oil in a large skillet over medium-high heat. Add the butter and melt. When all the butter is melted add the mushrooms and sautéed until soft, stirring often to avoid burning the butter, about 5 – 6 minutes. Transfer the mushrooms to a plate. Set aside.
Wipe the skillet out with a paper towel. Return to heat and warm some olive oil in skillet. Reduced the heat to medium and add the broccoli slices. Salt and pepper to taste. Sauté for 3 minutes; turn over and sauté the other side.
Return the mushrooms back to the skillet to heat through. Transfer the broccoli slices to serving plates. Spoon some mushrooms over each broccoli slice. Divide the feta cheese evenly over the slices. Season with breadcrumbs and fresh thyme.