Heat a large nonstick skillet over medium heat. Add 2 teaspoons oil to pan; swirl to coat. Add 2/3 cup shallots; cook 3 minutes. Add mushrooms; cook 4 minutes or until tender, stirring occasionally. Add wine, 1/4 teaspoon pepper, and salt; bring to a boil, scraping pan to loosen browned bits. Cook 2 minutes or until liquid almost evaporates, stirring occasionally. Remove mushroom mixture from pan; keep warm.
Return pan to medium heat. Add 1 teaspoon oil to pan; swirl to coat. Add remaining 1/3 cup shallots; sauté 5 minutes or until lightly browned. Remove shallots from pan; keep warm.
Preheat broiler to high.
Spread 2 teaspoons pesto over one side of each bread slice. Top each slice with about 2 tablespoons cheese. Broil 2 minutes or until cheese melts; keep warm.
Return pan to medium heat. Add remaining 1 teaspoon oil to pan; swirl to coat. Crack eggs into pan, and cook 4 minutes or until whites are set.
Top each bread slice with 2 tomato slices. Divide mushroom mixture evenly among bread slices, and top each serving with 1 egg. Sprinkle with remaining 1/4 teaspoon pepper, shallots, and basil.