Preheat oil in a deep-fryer to 350 degrees F.
Remove the gills and stems from the underside of the mushrooms and trim the edges off. Slice into 1/4-inch strips.
Measure flour into a pie plate. In another pie plate or bowl, combine panko, Parmigiano, parsley, red pepper flakes and salt and pepper.
Dredge the mushrooms in flour, followed by the egg wash and finally in the panko.
Working in batches, place the breaded mushroom slices in the hot oil and fry for 3 to 4 minutes, until golden brown. Drain on a paper towel lined sheet tray.