Combine 1 cup water and mung beans in a pot and bring to a boil; cook over medium heat until beans are very tender, about 30 minutes. Mash beans thoroughly, then stir in 1/2 cup sugar until well combined; set aside.

Mix together rice flour, remaining 1/4 cup water, 2 tablespoons sugar, and salt in a large bowl; knead just enough to form a ball. Divide dough into small pieces and roll into balls. Flatten each dough ball, then spoon 1 tablespoon mashed bean mixture into the middle of dough. Wrap dough around bean mixture and pinch edges to seal.

Heat oil in a deep fryer or large saucepan to 350 degrees F (175 degrees C). Fry balls in batches in hot oil until golden brown, 5 to 10 minutes per batch. Drain on paper towels and serve hot.