Steam the fresh broccoli florets until they’re just tender and then let them cool. Finely chop the broccoli into about ¼‑inch or smaller pieces for even texture. Place in a large bowl and mix with the shredded cheeses, Velveeta chunks, red pepper flakes and ¼ cup of the Panko crumbs and the 1 beaten egg.
Use your hands to form the mixture into balls (a rounded tablespoon per ball gives you around 12–14). Place them on a tray or plate and refrigerate for at least 30 minutes. Chilling helps them hold their shape during the breading and frying.
Set up three bowls: one with the flour, one with the two beaten eggs (plus a tablespoon of water if you like), and one with the remaining Panko crumbs. Heat oil in your pot to ~375 °F. Roll each chilled ball in flour, then in the egg mixture, then coat thoroughly in Panko. Fry a few at a time until they’re golden brown on all sides. Drain on paper towels.