Make a pea puree: In a small saucepan, heat 1 tablespoon of olive oil over medium heat. Add the shallot, season with a generous pinch of salt, and cook until tender, about 5 minutes. Add half of the peas and the chicken broth. Increase the heat to medium-high and cook the peas until tender, 2 to 3 minutes. Add the parsley and the tarragon. Transfer the mixture to a blender or food processor (or use a hand blender right in the pot) and puree it.

Crisp the prosciutto: Heat about a teaspoon of olive oil in a medium sauté pan. Add the prosciutto and cook, stirring occasionally, until the prosciutto is quite crispy, 4 to 5 minutes.

To serve: Cook the pasta in well-salted boiling water. If using fresh pasta, add it and the other half of the peas to the pot at the same time and cook until the pasta is al dente. If using dried, start the pasta first and add the peas for the last 2 minutes of cooking. Reserve a cup or so of the pasta water and then drain the pasta and peas. Return the pasta to the pot it cooked in, add the pea puree and toss to coat, adding a little of the reserved pasta water if it looks dry. Divide the pasta among warm bowls and top the pasta with the prosciutto. Drizzle a teaspoon of olive oil over each bowl and serve immediately.