Preheat oven to 400℉. In a medium saucepan, boil enough water to prepare the baking dish (see step 12 below).
Heat the butter in a large skillet over moderately high heat.
Add the mushrooms, shallots, garlic, tarragon, salt and pepper and cook until mushrooms are tender and the liquid has evaporated, about 8 minutes (this will cook down to about 1 1/4 cups mushrooms or 12 tablespoons).
Remove from the heat and carefully stir in the brandy. Return to the heat and cook for 1 minute. Set aside.
Line a baking dish or roasting pan with a small clean dish towel (this will keep the ramekins from sliding).
Lightly grease the bottom and sides of the ramekins.
Spoon 2-3 Tablespoons of the mushroom mixture into the ramekins. With a spoon, make a small indentation in the middle of the mixture where the egg yolk will go.
Break 1 or 2 eggs onto the mushroom mixture.
Sprinkle the egg(s) with a pinch of salt and pepper.
Stir together the whipping cream and grated cheese.
Top the eggs with 1-2 Tablespoons of the cream mixture.
Place the ramekins in the baking pan and fill with enough very hot water to reach halfway up the sides of the ramekins. (Sometimes it is easier to add the hot water when the pan is on the oven shelf.)
Gently transfer the baking dish with the ramekins to the middle rack of the oven.
Cover the baking dish with foil and bake until the whites are opaque and firm but the yolks are still runny, about 13-15 minutes or until cooked to your liking.
Carefully remove the ramekins using oven mitts.
Allow to set for one minute then sprinkle with chopped parsley or chives.
Serve with your favorite toasted bread.