How to Make It
Preheat the oven to 250°. Line a baking sheet with a silicone baking mat and arrange the prosciutto in an even layer on top. Bake for about 2 hours in the center of the oven, flipping once, until dehydrated and very crisp. Transfer to paper towels to drain and cool, then break into pieces.
Meanwhile, fill a large bowl with ice water. In a large saucepan of salted boiling water, blanch the peas until just tender, about 2 minutes. Using a slotted spoon, transfer the peas to the ice water to cool. Reserve 1 3/4 cups of the cooking water.
Drain the peas and add to a blender. With the machine on, add the reserved cooking water in a thin stream and puree until very smooth. (You may not need all of the reserved water.) Blend in simple syrup to taste. Strain the soup through a fine sieve and season with salt and pepper. Refrigerate until chilled, about 30 minutes. Ladle the soup into bowls, top with the prosciutto crisps and a drizzle of olive oil and serve.
For faster prosciutto crisps: Preheat the oven to 350°. On a parchment paper–lined baking sheet, arrange the slices of prosciutto in a single layer and bake until crisp, about 17 minutes. Transfer to paper towels to drain and cool, then break into large pieces.