2 large eggs
4 tablespoons extra-virgin olive oil, plus more for serving
2 large slices of rustic bread, such as ciabatta
1/2 pound mixed cleaned wild mushrooms (such as hen of the woods, black trumpet, or oyster)
1 medium clove garlic, minced on a microplane grater (about 1 teaspoon)
1 small shallot, finely chopped (about 1 tablespoon)
1 teaspoon picked thyme leaves
1 teaspoon juice from 1 lemon
Kosher salt and freshly ground black pepper