Heat about 1 tablespoon butter (per potato patty) over medium heat on griddle or large skillet. Add 1 cup shredded potatoes to form a patty. Make 12 patties total. Season with salt and pepper. Cook until golden brown on bottom; turn over and brown on second side. Keep patties warm.
Into a large skillet, heat olive oil and 2 tablespoons butter. Stir in spinach; sauté until wilted. Season with salt; keep warm.
Poach eggs in water (acidified with 1 teaspoon vinegar per quart of simmering water) until whites are set (completely coagulated and firm) and yolks begin to thicken (no longer runny, but not hard).
For each portion, place one hash brown patty on a plate. Top with 1/4 cup spinach, a slice of prosciutto, poached egg and 1/4 cup hollandaise sauce. Serve immediately.