In a medium saucepan, melt the 4 tablespoons of butter. Whisk in the flour and cook over moderate heat for 1 minute. Whisk in the milk and cook over moderately low heat until smooth and very thick, about 2 minutes. Stir in the salt and cayenne. Off the heat, whisk in the egg yolks. Let cool slightly. Transfer to a large bowl and stir in the chèvre and mushrooms.