For this dish, you will need a pan with a lid, and a glass lid would be ideal so you can keep an eye on the eggs.
Heat a non stick pan on medium heat and melt the butter, then add the mushrooms, onion, garlic, and thyme.
Saute to soften the mushrooms and release any water and soften the onion, season with salt and pepper as desired. Saute until most of the water is evaporated.
Sprinkle the cheese evenly over the mushrooms, crack the eggs on top, season with salt and pepper as desired. If all the water has cooked off, simply add a tablespoon or two of water on the side. (I did not use cheese this time.)
Cover with the lid to poach the eggs, whites are set and the yolks are runny or to your liking.
Season with more pepper if desired.
Serve with toast, enjoy.