1.) In a large bowl, beat together 6 eggs and cream. Add bread cubes. Stir until bread is coated with egg mixture. Set aside.
2.) Sauté mushrooms in butter until all water is released. Add Parmesan cheese, parsley and lemon juice. Season with season with salt and pepper. Add to bread mix.
3.) Divide mixture into six buttered muffin tins.
4.) Make well in center of each, and bake at 350° F until an internal temperature of 160° F is reached and the bread mixture is just set.
5.) Remove from tin and place on baking sheet. Add one egg into each well and return to oven until egg whites are firm and completely coagulated, and yolks are beginning to thicken but are still soft.
6.) Assemble by plating mushroom bread pudding in bowl and pour hot mushroom broth just prior to service, or serve mushroom broth in small pitcher to pour at tableside.