To make the ice-cream, place sugar and ? cup cold water in a saucepan and stir over low heat until sugar dissolves. Simmer for 2 minutes, then remove from heat. Mix 1 tablespoon of syrup with the tea, then add to syrup in saucepan. Cool completely.
Whip the thickened cream. Mix with thin cream and tea syrup. Pour mixture into a shallow container and freeze until frozen at edges. Remove from freezer and beat with an electric beater. Pour back into container and re-freeze. Repeat 2 or 3 times. (Alternatively use an ice-cream maker following manufacturer's directions.)
Sift ½ cup plain flour and the cornflour into a bowl. Add a pinch of salt, the egg and ½ cup cold water. Whisk to a smooth batter with a hand whisk, then set aside.
Half fill a deep-fryer or large heavy-based saucepan with oil and heat to 190°C. (If you lack a deep-fryer thermometer, test a cube of bread - it will turn golden in 30 seconds if the oil is hot enough.) Peel bananas, cut each in half across the centre and toss in remaining flour to coat. Working in batches, dip each piece of banana in batter, then carefully drop in the hot oil. Fry for 1-2 minutes or until golden brown (you may need to turn the pieces over halfway through cooking). Dust with icing sugar and serve hot with ice-cream.