Heat the oil in a pan over medium heat, add the onions and mushrooms and cook until the mushrooms release their moisture, it evaporates and the mushrooms start to caramelize, about 15 minutes.
Add the garlic thyme and brussels sprouts, saute until tender, about 5 minutes, let sit until the bottoms start to caramelize, about 5 minutes, mix everything up, let sit for 5 minutes and mix it all up.Season with salt and pepper to taste.Create small wells in the hash, add the eggs and cook until the desired level of doneness.