1. Fill a large saucepan a quarter full of water and heat over high heat.
2. Select a Pyrex or metal bowl that fits over top of the saucepan, making sure only the bottom of the bowl touches the water.
3. When water is simmering, place bowl in saucepan. Reduce to medium heat, place butter in bowl and let melt.
4. Squeeze lemon half over the bowl, then add water and mustard. Season generously with pepper and add a pinch of salt. Stir in egg yolks all at once and whisk continuously for 5–6 minutes until sauce thickens slightly. Remove bowl from saucepan and set aside.
5. Fill the same saucepan with water and bring to a boil over high heat.
6. Meanwhile, pour vinegar onto a large plate. Gently break eggs into vinegar and let sit 5 minutes.
7. Using a slotted spoon, stir boiling water in a circular motion until it is swirling smoothly around. Carefully pour plate contents into saucepan, reduce heat to medium-high, and cook eggs for 3 minutes.
8. Using the same slotted spoon, remove eggs from saucepan and place them on a plate lined with a clean cloth.
9. Toast bread.
10. Divide toasted bread slices onto 4 plates. Garnish toast with arugula, a slice of prosciutto, a poached egg and hollandaise sauce. Sprinkle with grated parmesan and crushed pepper. Serve.

This recipe is best prepared just before serving.