Combine mayonnaise and pesto in a small bowl; refrigerate.
Combine rice, cheese, eggs, prosciutto and pepper in a large bowl and mix well. Shape into 20 patties, about 1 ½-inches in diameter. Heat oil in a large nonstick skillet over medium heat. Sauté half the patties 2 to 3 minutes on each side or until golden brown; remove from pan. Cook remaining patties, using additional oil, if needed.