Place half of the English muffin pieces in 2 lightly greased 10-ounce custard cups. Sprinkle with ham. Top with asparagus and cheese. Add the remaining English muffin pieces.
In a small bowl combine the sour cream, onion, mustard, caraway seeds (if desired), and black pepper. Gradually stir in milk and beaten egg. Pour evenly over the layers in the custard cups. Lightly press with the back of a spoon to thoroughly moisten English muffin pieces. Cover and refrigerate for 2 to 24 hours.