Step 1

Preheat oven to 425 degrees. Toast flatbread over the flame of a gas burner or under broiler. Tear into 2-inch pieces; set aside.

Step 2

Heat oil and garlic in a large ovenproof skillet over medium heat, stirring, until garlic is fragrant but not browned, about 1 minute. Increase heat to medium-high, add mushrooms, and cook, stirring occasionally, until tender and golden brown, about 10 minutes. Season with salt and red-pepper flakes. Add wine, scraping up browned bits. Stir in flatbread and thyme; remove from heat.

Step 3

Make 4 shallow wells in mushroom mixture and crack 1 egg into each well. Transfer to oven and cook until egg whites have just set, about 4 minutes. Season eggs with salt and red-pepper flakes.