Cook pasta according to package directions. Drain, reserving ½ cup pasta water.
Heat olive oil and prosciutto in a large skillet over medium. Cook, stirring, until prosciutto is crisp, about 4 minutes; transfer to a plate using tongs. Add garlic to skillet and cook 30 seconds. Add spinach, salt, pepper, and lemon zest and cook, tossing, until spinach is wilted, 2 minutes.
Add pasta, reserved pasta water, and half the cheese to skillet. Cook, tossing, until sauce thickens and coats the pasta, about 1 minute. Top with prosciutto and remaining cheese.