Preheat oven to 350 degrees F. Cook the bacon in a medium skillet, over medium-high heat until just crisp. Transfer to paper towels to drain. Discard all but 2 tablespoons of the fat in the pan. Add 2 tablespoons butter, the mushrooms and 1/2 teaspoon salt. Cook, stirring over medium heat, until the mushroom juices evaporate, about 7 to 10 minutes. Add the shallots and cook until tender and mushrooms are golden, about 3 minutes more. Add the garlic and parsley. Remove from heat, cool slightly.
Brush a 9-inch glass or ceramic pie pan with the remaining butter and sprinkle evenly with the grated Parmesan. Place pan on a baking sheet. Scatter half the Gruyere into the pan and crumble the bacon on top. Add 3/4 of the mushrooms, then the remaining cheese.
Whisk the cream, eggs and yolks in large glass measuring cup. Season with salt, pepper, and nutmeg to taste. Pour the custard over the fillings. Top with remaining mushroom mixture and sprinkle with chives.
Bake until the quiche is just set in the center, about 40 to 50 minutes. Cool completely on a rack before serving.