Open-Faced Sandwiches with Mushrooms and Fried Eggs

Preheat a large nonstick skillet over medium-high heat. Add 2 teaspoons of oil to the skillet; stir to coat. Add 2/3 cup shallots; cook for 3 minutes. Add the mushrooms; cook for 4 minutes or until tender, stirring occasionally. Add the wine, 1/4 teaspoon pepper, and salt; bring to a boil, scraping the pan to prevent browning. Cook for 2 minutes or until the liquid is almost evaporated, stirring occasionally. Remove the mushroom mixture from the pan; keep warm.Put the skillet on medium heat. Add 1 teaspoon of oil to the skillet; stir to coat. Add the remaining 1/3 cup shallots; simmer for 5 minutes or until lightly browned. Remove the shallots from the pan; keep warm.Preheat the grill to high heat.Spread 2 teaspoons of pesto on one side of each slice of bread. Sprinkle each slice with about 2 tablespoons of cheese. Bake for 2 minutes or until the cheese is melted; keep warm.Put the skillet on medium heat. Add the remaining 1 teaspoon of oil to the pan; stir to coat. Add the eggs to the skillet and cook for 4 minutes or until the whites are set.Place 2 slices of tomato on each slice of bread. Spread the mushroom mixture evenly over the bread slices and brush each serving with 1 egg. Sprinkle with the remaining 1/4 teaspoon pepper, shallots, and basil.