Steam broccoli until slightly softened. Let cool.
Finely chop broccoli. You want the pieces to be 1/4-inch or less. Place in a large bowl.
Add cheese, red pepper flakes, 1/4 cup Panko crumbs, and 1 egg to bowl with broccoli. Stir well.
Use your hands to shape mixture into balls using about a rounded tablespoon for each ball. You should get 12-14.
Place balls on a plate and refrigerate for at least 30 minutes.
Place flour in a bowl.
Place the 2 lightly beaten eggs in a second bowl and mix in 1 tablespoon water.
Place remaining 2 cups Panko crumbs in a third bowl.
Pour about 2 inches of oil in a Dutch oven or heavy pot. Heat oil to 375 degrees. For best results use a thermometer.
Remove broccoli balls from refrigerator, coat in flour, dip in egg mixture, and then coat in Panko crumbs.
Fry about 4 balls at a time cooking until golden brown on all sides.
Drain on a paper towel-lined plate.
