Whisk egg with 1 tbsp. water. Lightly brush edges of discos with some of the egg wash. Spoon 2 tsp. of cream cheese and 1 tsp. of guava paste into center of each disco. Fold over and pinch edges together to seal.
Meanwhile, pour enough oil into large saucepan to come 3 inches up side; heat over medium heat until instant-read thermometer reads 350°F.
Working in 3 batches, carefully drop mini empanadas into hot oil; fry for 2 to 3 minutes or until golden brown all over. Using slotted spoon, transfer empanadas to paper towel–lined tray. Let cool to room temperature; dust with icing sugar.
If you don’t have an instant-read thermometer, check the heat of the oil by dipping the handle of wooden spoon into oil; if consistent and steady bubbles form, oil is hot enough. If bubbles are too few, oil is not hot enough; and, if too vigorous, oil is too hot.