Fill a deep skillet or frying pan with enough oil to reach about 1 inch up the sides. Warm over medium heat until the oil is hot enough to make a bread cube sizzle.
In the meantime, put the flour into a mixing bowl large enough to accommodate a stalk. Pour in the sparkling water and mix with a whisk until the mixture is blended and smooth. It should have the consistency of beaten egg and easily coat a spoon. If the mixture is too thick, thin it with a little more of the water.
Cut off any discolored tip from the base of the broccoli rabe stalk but leave their leaves and crowns intact. Wash them and pat them dry thoroughly with a paper towel. Dip each of the stalks into the batter just before you are ready to cook them, using the stem as a handle. Slip them carefully into the pan to avoid splattering. Put only as many stalks into the oil as can fit easily without crowding. Fry until they have a golden cast, about 6 minutes altogether, turning them over midway during cooking.
Use a skimmer to retrieve them when they are done. Transfer to paper towels, taking care to avoid crushing them. Sprinkle them with salt immediately to ensure that it will stick and keep them crisp. Serve at once.