Beat the eggs in a mixing bowl with a wire whisk until smooth. Stir in the grated and diced courgettes, then season with salt, pepper, Parmigiano Reggiano and basil. Add the flour a little at a time, stirring between additions until no dry lumps remain.
Heat the oil in a large frying pan to 190 degrees C.
Drop the batter into the hot oil by the spoonful a few at a time making sure not to overcrowd the oil. Cook until the fritters are golden brown on each side, about 4 minutes. Drain on a kitchen towel-lined plate before serving.
If the courgettes are a little wet, squeeze and pat dry with kitchen towels.