In a large heavy skillet, heat enough oil to come 1/4-inch up the sides of the pan to 350°F. In batches, lightly coat the vegetables in the batter. Carefully transfer to the hot oil and fry, turning to brown evenly, being careful not to add too many pieces at once.
Transfer to paper towels to drain, and season with salt. Sprinkle lightly with the remaining grated Parmesan and the minced prosciutto. Repeat with the remaining ingredients and serve.