Leeks and Mushrooms on Cheesy Toasts with Fried Eggs

Heat 1 tablespoon oil in a large skillet over medium heat; add the leeks and 1/2 teaspoon each salt and pepper and cook, covered, stirring occasionally, until tender, 7 to 8 minutes; transfer to a plate. Heat 2 tablespoons oil in the skillet (no need to clean) over medium-high heat. Add the mushrooms and cook, tossing occasionally, until golden brown and tender, 3 to 5 minutes. Add the garlic to the skillet and cook, stirring, for 1 minute. Reduce heat to low, return the leeks to the pan along with the spinach and cook, folding together until the spinach begins to wilt, about 2 minutes; remove from heat and stir in the lemon juice and tarragon. Meanwhile, heat the broiler. Heat the remaining tablespoon oil in a large nonstick skillet over medium heat. Crack the eggs into the pan and cook, covered, to desired doneness, about 2 minutes for slightly runny yolks. Place the bread on a foil-lined broiler-proof baking sheet. Sprinkle with the cheese. Broil until the cheese starts to brown, 1 to 2 minutes. Serve the eggs and cheesy toast with the leek mixture.