Whisk together flour, brown sugar, baking powder, and salt in a large bowl, then add water and egg and whisk until batter is smooth.
Peel plantains and cut on a slight diagonal into 1/2-inch pieces. Stir into batter to coat well.
Stir together granulated sugar and cinnamon in a shallow bowl.
Heat 1/2 inch oil in a 10-inch heavy skillet over moderate heat until thermometer registers 340°F. (See cooks' note, below.) Fry plantain slices in batches of 6 (don't crowd them) until bottoms are golden, about 45 seconds, then turn over and fry until other side of each is golden, 30 to 45 seconds more.
Transfer with a slotted spoon to paper towels to drain.
While still warm, toss each batch in sugar mixture until coated, then transfer to a platter.
Serve hot or warm.
To take the temperature of a shallow amount of oil with a metal flat-framed deep-fat thermometer, put bulb of thermometer in skillet and turn thermometer facedown, resting other end (not plastic handle) against rim of skillet. Check temperature frequently.
