Preheat deep fryer to 375 degrees F.
Beat the egg and milk together, add 1/2 cup of the flour and whisk together well to blend. Mix the remaining cup of flour with the salt, pepper, and Cajun seasoning. Dip the frozen broccoli florets into the batter, then hold it up over the bowl, allowing it to drain well. Toss in the seasoned flour. Drop the broccoli carefully into hot oil and fry until golden brown, taking care not to overcrowd.
Good just as they are, or serve with ranch dressing, cocktail sauce, comeback sauce, remoulade sauce, or a spicy honey mustard sauce.
I use frozen broccoli for deep frying mainly because I think they work better and it's a freezer staple in my house. If you want to use fresh, raw broccoli, you'll first need to cut it into small florets, then blanch them about a minute in boiling water to partially cook them. Transfer to an ice water bath immediately to stop the cooking process, the let them dry thoroughly before proceeding.