Bring the whole plantains with enough water to cover to a boil; reduce heat to medium-low, cover, and simmer until the plantains are tender and the skins have cracked, about 45 minutes. Drain and set aside until cool enough to handle.
Heat oil in a deep-fryer or large saucepan to 375 degrees F (190 degrees C).
Peel the plantains and mash with the salt and pepper until no lumps remain. Roll the mashed plantain into 3 tablespoon size balls around a cube of cheese. Cook the plantain balls in the hot oil in batches until they are golden brown and float to the surface, about 5 minutes. Move the plantain balls around as they cook to ensure they brown evenly. Drain the cooked plantain balls on a paper towel-lined plate to cool slightly.