Use a vegetable peeler to peel strips away from a whole cucumber. Choose the best 12 strips that are equal length (3 strips used per rollup). The total strips you use should equal the weight listed in the ingredients. Lay the cucumber strips flat on a piece of paper towel and press a second paper towel over the strips to absorb excess liquid.
Arrange each rollup by laying 3 cucumber strips vertically. Overlap each strip about halfway over the previous one. Then, lay a slice of prosciutto over each portion of layered cucumber strips. Lay a ½-ounce slice of fresh mozzarella cheese at the base of each strip of prosciutto. Follow up by placing a single basil leaf on top of each slice of cheese.
Drizzle the EVOO gently over each portion of mozzarella cheese. One by one, roll each rollup by wrapping the end of cucumber and prosciutto nearest you over the mozzarella cheese slice. Then, tightly roll up each portion, tucking the remaining end of cucumber and prosciutto under each rollup. Eat right away as a snack or part of an easy lunch.