Bring a large pot of salted water to boil.
In a large skillet, heat a thin layer of olive oil over a medium-high flame. Once the pan is hot, add the prosciutto in an even layer (you'll have to do this in batched) and fry until crispy on both sides, about 2 minutes total. Remove to a plate lined with paper towels, and repeat with the remaining proscuitto. Once cool enough to touch, crumble the prosciutto into rustic bite-sized pieces.
Add the zucchini to the pan, along with any necessary additional olive oil - the pan will need a light coating. Pan-fry in an even layer until golden brown on both sides, stirring once or twice. Season with salt and set aside. 4. Cook the pasta according to package directions until al dente.
Meanwhile, whisk together the egg and half the cheese. Temper the eggs: slowly whisk in 2 tablespoons of pasta cooking water to thin the egg mixture and raise the temperature. Drain the pasta and immediately add to the egg mixture, tossing until the sauce clings to every strand. Add the zucchini, prosciutto, and half the chives and toss to combine. Taste for seasoning and add more salt as necessary.
Divide the pasta among four plates, or transfer to a serving platter and garnish with the remaining cheese and chives.