Steam the broccoli florets until slightly tender but not soft. Remove from heat and allow to cool completely.

Finely chop the broccoli into pieces no larger than ¼ inch. Transfer the chopped broccoli to a large mixing bowl.

Add the Colby cheese, cheddar cheese, Velveeta cubes, crushed red pepper flakes, ¼ cup of the panko breadcrumbs, and the lightly beaten egg to the bowl. Stir until the mixture is thoroughly combined.

Using clean hands, shape the mixture into balls, using approximately one rounded tablespoon per portion. You should obtain approximately 12–14 balls.

Place the formed broccoli cheese balls onto a plate or tray and refrigerate for at least 30 minutes to allow them to firm.

Prepare a breading station: place the flour in one shallow bowl, the two lightly beaten eggs mixed with 1 tablespoon of water in a second bowl, and the remaining 2 cups of panko breadcrumbs in a third bowl.

In a Dutch oven or heavy-bottomed pot, heat approximately 2 inches of oil to 375°F (190°C). Use a thermometer to maintain an accurate temperature.

Remove the chilled broccoli cheese balls from the refrigerator. Coat each ball first in flour, then dip into the egg mixture, and finally coat evenly with panko breadcrumbs.

Fry the balls in batches of 3–4 at a time, turning gently, until golden brown on all sides.

Remove the fried balls from the oil and drain on a paper towel–lined plate. Serve warm.