Using a soft bristle pastry brush, brush a sheet of phyllo with the melted butter or margarine.
Place a second sheet on top and smooth it with your hands.
Repeat until you have a stack of all six sheets. Brush the top with butter.
Place on a cutting board. Using a sharp knife, cut the stacked sheet into 6 squares (approximately 5 inches).
Lightly brush butter into a muffin tin with 6 cups.
Push the squares into the muffin cups, buttered side up. Mold them to fit smoothly into the cups with the corners sticking up around the edge.
Distribute the cheese evenly in the bottom of the pastry cups.
Sprinkle the ham over the cheese.
Carefully break an egg into each cup.
Season with freshly ground pepper and salt.
Place in the center of a pre-heated 350ºF. oven.
Bake for 12 minutes for soft eggs and 15 minutes for hard eggs.
Serve hot. Top each with a small sprig of parsley.