Prepare the vegetables: Peel the potatoes and cut into large chunks, about 1.5-inch (4 cm). Trim green beans (only the stem ends) and cut in half. Cut cherry tomatoes in half. Shave pecorino cheese into strips with a vegetable peeler.
Cook the eggs and potatoes (I'm cooking the eggs and potatoes in the same pot but you can cook them separately if you wish): Add the potatoes into a large pot and pour in boiling water. Salt the water generously, bring to a boil and cook over medium-low heat for 8 minutes. Add the eggs to the pot: use a large spoon to lower the eggs into the boiling water, bring water to a boil, and cook for 9 minutes. When the time is up, transfer the eggs to an ice bath (a bowl filled with cold water and ice cubes or just very cold water) and crack their shells immediately with a spoon (this helps to peel them). Drain the potatoes and leave them in the pot.
Cook the green beans: add the green beans into a medium pot, pour in boiling water, salt the water. Cook the green beans until tender, about 2-3 minutes for thin green beans or 6-7 minutes for thick beans.
Make the pesto: Add all the ingredients for the pesto sauce except the olive oil into a food processor. Pulse until a thick paste forms (scrape the ingredients from the sides of the bowl from time to time). Add the olive oil and pulse the sauce at the lowest speed until well combined. Season to taste with salt and pepper. Add the pesto sauce to warm potatoes and toss to coat.
Cook prosciutto: Heat 1 tablespoon of oil in a non-stick pan over medium heat. When hot, add prosciutto slices and cook (in batches) until browned and crispy on both sides. Transfer to a plate lined with paper towels to drain excess fat.
Assemble the salad: Peel the eggs and cut into large chunks. Add the eggs, green beans, cherry tomatoes, and pecorino cheese to the pot with potatoes. Toss everything together, season with salt and pepper if necessary. Divide the salad onto plates and top with crispy prosciutto broken into smaller pieces.