Preheat the oven to 400°F. In a large skillet, heat the olive oil and butter over medium heat. Add the sliced leeks and season lightly with salt and pepper. Cook the leeks over medium heat, stirring often until they become translucent and soft, about 7 to 10 minutes. Add the asparagus pieces to the pan and cook for 2 minutes more. Remove the pan from the heat and set aside.
Using six 8 oz. ramekins or skillets, divide the vegetable mixture into portions. Crack two eggs in each ramekin. Add 2 tbsp. of heavy cream to each portion and top with the thyme, tarragon and salt and pepper to taste.
Sprinkle 1 tbsp. Parmesan cheese on top of each portion and bake uncovered on a sheet tray in the oven until the eggs are cooked to desired temperature, about 8 to 10 minutes for a medium runny egg (make sure whites are thoroughly cooked). Serve hot with toasted bread for garnish.