1. Steam the Broccoli:
   Steam broccoli until slightly softened, then let it cool. Finely chop broccoli into pieces about ¼-inch or smaller.
2. Prepare the Mixture:
   In a large bowl, combine the chopped broccoli, Colby cheese, cheddar cheese, Velveeta, red pepper flakes, ¼ cup Panko crumbs, and 1 beaten egg. Mix well.
3. Shape the Balls:
   Use about a rounded tablespoon of mixture for each ball, shaping it by hand. You should get around 12-14 balls. Place on a plate and refrigerate for at least 30 minutes.
4. Prepare Coating:
   Set up three bowls for coating. Add flour to the first bowl. In the second bowl, mix the 2 lightly beaten eggs with 1 tablespoon of water. Place the remaining 2 cups of Panko crumbs in the third bowl.
5. Coat the Balls:
   Remove broccoli balls from the refrigerator, coat each one in flour, then dip in the egg mixture, and finally roll in Panko crumbs.
6. Fry the Balls:
   Pour about 2 inches of oil in a Dutch oven or heavy pot and heat to 375°F (190°C). Fry about 4 balls at a time, cooking until golden brown on all sides. Remove and drain on a paper towel-lined plate.

Enjoy these crispy, gooey, and cheesy broccoli balls as a tasty appetizer!