3/4 lb mixed mushrooms, (oyster, shitake, cremini) finely chopped
1/4 cup finely chopped shallot
2 tablespoons unsalted butter
1/2 teaspoon salt
1/4 teaspoon black pepper
2 tablespoons creme fraiche or sour cream
1 tablespoon finely chopped fresh tarragon
4 thin slices of Virginia ham or prosciutto (optional)
12 large eggs
12 individual brioche
Cream

Garnish: fresh tarragon leaves
Special equipment: a muffin tin with 12 (1/2-cup) muffin cups

Preheat oven to 375°F.

Prepare mushrooms:
Cook mushrooms and shallot in butter with salt and pepper in a large heavy skillet over moderately high heat, stirring, until mushrooms are tender and liquid they give off is evaporated, about 10 minutes. Remove from heat and stir in creme fraiche and tarragon.

Assemble and bake:
Cut top off brioche and hollow out cup. Butter inside of brioche shell and bottom of the top piece. Place brioche bottoms in muffin tin.

Cut up the ham, if using meat, and line the bread. Divide mushrooms among brioche cups and spread in the bottom. Break 1 egg into each. Top each egg with 2 tsp. cream. Bake in middle of oven until whites are cooked but yolks are still runny, about 18 minutes. Toast brioche tops during last five minutes of baking. Season eggs with salt and pepper and remove from muffin cups carefully, using 2 spoons or small spatulas.