In a cast iron pan, heat oil over medium heat.
Add butternut squash and sauté until slightly browned, stirring occasionally, about 5 minutes.
Add onion and mushrooms and sauté until mushrooms are browned and onions translucent, about 5 minutes. Season with salt and pepper, to taste.
Stir in thyme and sage and cook until fragrant, about 1 minute.
Turn down heat to low. Using the back of a wooden spoon, create 4 small wells in the hash. Crack an egg into each of the wells.
Cover the pan and cook until egg whites are set, about 7-10 minutes.
Season with additional salt and pepper, to taste.
Optional: garnish with extra chopped herbs.