In a large pan, add 1 tablespoon butter and 1 tablespoon olive oil; melt over medium-high heat.
Stir in half of the minced garlic and cook for 20 seconds.
Add mushrooms and cook for 8 minutes, stirring frequently, until mushrooms are caramelized and tender.
Transfer mushrooms to a plate and keep covered.
Heat remaining butter and olive oil in the pan.
Add diced onions and cook for 3 minutes.
Add remaining garlic and cook for 10 seconds.
Stir in rice; add thyme, salt, and pepper, and cook for 30 seconds.
Add 1/2 cup broth and, using a wooden spoon, scrape up all the browned bits from the bottom of the pan.
Stir in remaining broth and bring to a simmer.
Reduce heat to Low; cover with a lid and cook for 15 to 17 minutes, or until liquid has evaporated.
Remove from heat and stir in the previously prepared mushrooms and green onions; cover and set aside for 10 minutes.
Fluff rice with a fork and serve.