Preheat oven to 400° with a rack in the center. Line a 12- by 17-in. rimmed baking sheet with parchment paper and coat lightly with oil.
Put mushrooms and leeks on baking sheet. Sprinkle with oil, 1/2 tsp. each salt and pepper, and the thyme and toss to coat. Spread vegetables in an even layer. Bake until softened and starting to brown, about 10 minutes.
Remove baking sheet from oven but leave oven on. Make 8 evenly spaced hollows in vegetables and crack an egg into each. Sprinkle with goat cheese and salt and pepper to taste.
Return baking sheet to oven and bake until egg whites are set but shiny and yolks are still runny, 8 to 10 minutes; jiggle baking sheet to check softness of yolks.