In a nonstick skillet over medium-high heat, melt butter and add onions & mushrooms.
Season with thyme, salt & pepper and cook, stirring, until onions are soft and liquid has been released and absorbed from mushrooms, approximately 5 minutes.
Preheat oven to 375°F (190°C) and grease a 6-cup muffin tray.
With a cup that has an opening that is the same size as the bottom of the muffin cups, cut the bread into rounds, then place a bread round into the bottom of each muffin cup and divide cooked onion mixture evenly among the six compartments.
Sprinkle 1⁄2 cup of the cheese evenly in the same way. Gently press down on cheese mixture and crack one egg into each compartment.
Bake in preheated oven for 15 minutes.
Take out of oven and sprinkle the remaining cheese on each egg and continue to bake for 3 minutes.