Peel the potatoes and cut into 1-inch cubes. Place the potatoes in a large pot filled with salted water. Bring the water to a boil, lower to a simmer and let potatoes cook until cooked through, about 8 minutes. When the potatoes are cooked, drain and pass through a food mill. Add the chopped garlic, parsley, salt, pepper and eggs to the potato puree. Stir in a drizzle extra-virgin olive oil with a wooden spoon until seasoning is spread evenly throughout the puree. Allow the puree to cool.
While potato mixture cools, fill a deep pot no more than halfway full with extra-virgin olive oil. Over a medium-high heat, heat the oil to 360 to 365 degrees F. The oil should remain at or around this temperature throughout the frying process. Using your palms, shape chunks of the potato puree into oval-shaped croquettes, about 2 inches in length. When the oil reaches 350 degrees F, begin frying the fritters in batches of 6 until golden brown. Use a slotted spoon or spider to gently drop the fritters into the water, being careful not to splatter the hot oil. Using the same spoon or spider, remove the finished cazilli to a serving plate. Repeat the frying process until all the potato mixture has been used.
Garnish with parsley and cheese.