Heat ½ Tbsp oil and butter in a large skillet over medium-high heat. Add mushrooms and cook 5 minutes or until lightly browned and beginning to soften. Season with ¼ tsp each of salt and pepper. Remove mushroom mixture from skillet, and wipe skillet clean.
Heat remaining 1 Tbsp oil in skillet over medium heat. Add onion and garlic; cook 5 minutes or until onion is tender. Add cauliflower rice, broth, and thyme, and bring to a simmer. Cover, and cook 10 minutes or until cauliflower is crisp-tender.
Remove lid and stir. Cook, uncovered, 10 to 12 minutes or until there is no more liquid. Gently stir in mushrooms, cheese, and remaining ¼ tsp each salt and pepper. Serve immediately.