In a medium bowl, mash bananas, vanilla extract, and nutmeg with a fork until fully pureed; you should have about 1 1/2 cups mashed banana.
Stir in flour followed by the milk (or water) until a thick and slightly lumpy batter forms (exactly how thick varies from cook to cook, but a general goal is a batter that's thicker than pancake batter, spreadable, but not free-flowing); add additional liquid 1 tablespoon at a time, if needed, to achieve the desired texture.
Heat oil in a medium sauté or frying pan over medium-high heat until shimmering. Working in batches to avoid crowding the pan, carefully add roughly 2-tablespoon-sized spoonfuls of banana batter to oil, making sure to release them close to oil to prevent splashing, and fry until golden on first side, 1 to 2 minutes. Flip and fry until golden on second side side, 1 to 2 minutes longer. Transfer to paper towels to drain and repeat with remaining batter.
In a small bowl, stir together sugar and cinnamon. Transfer fritters to a serving dish and sprinkle cinnamon-sugar all over hot fritters. Serve hot or at room temperature as a side dish with a meal or as a quick and tasty snack.