Preheat oven to 375˚. Lightly grease four 4-ounce ramekins.
Heat 1 tablespoon butter in a large sauté pan over medium heat and add leeks and mushrooms.
Sprinkle with salt and cook for 3–4 minutes, until mushrooms are softened.
Divide leeks and mushrooms among prepared ramekins.
Top each with about a tablespoon or 2 of heavy cream and 2 tablespoons cheese.
Crack one egg into a small cup and carefully pour egg over vegetables and cheese in one ramekin, being careful not to break yolk.
Repeat with remaining eggs and ramekins.
Arrange ramekins in a baking dish or glass casserole.
Pour boiling or very hot water into dish, halfway up ramekin sides. Carefully transfer dish to oven and bake 20–25 minutes (check at 15 minutes), until eggs are set but yolks are still tender.
Remove from oven and shower with fresh tarragon or parsley. Serve immediately.