Preheat oven to 350 degrees F. Prepare an ice bath for asparagus (You want to do this to end the cooking when you pull the asparagus out of the hot water).
In a large pot add water and bring to a boil. Add about a tablespoon of sea salt. Add asparagus until crisp-tender. About 3 minutes.
Drain. Transfer asparagus to your prepared ice bath. Let sit for 5 minutes then put onto a platter.
Heat 3 tablespoons olive oil in large sauté pan over medium heat. Add chopped anchovies and garlic and sauté until garlic is golden, about 1 minute. Add asparagus and prosciutto and sauté until heated through, about 3 minutes.
Add fresh lemon juice and remaining 1/2 cup olive oil and simmer 1 minute. Remove from heat.
Add salt and pepper then divide greens among 6 plates. Top with asparagus mixture equally. Pour dressing from skillet over each serving. Serve warm or cold