Using a sharp knife, remove the pits from the peaches by cutting a hole in the center of each peach, taking care to keep the rest of it intact.
Stir together the flour, sugar and salt. Add the egg and the beer, and whisk until just combined. If the batter is too thick you can add some additional beer.
Dunk the peaches in the batter, coating each fruit. Heat the oil until it reaches 375º. Transfer two peaches into the oil gently so as not to burn yourself. In just a few seconds (5-10) the peaches should be golden brown. Remove them from the oil and drain them on paper towels. Repeat with remaining peaches, frying them two at a time.
For the topping, mix together the cinnamon and sugar, and sprinkle some over each side of the doughnuts. Let the doughnuts cool slightly before serving.