Broccoli is one of my favourite vegetables. I generally cut it into bite size pieces, and blanch it before stir frying (nice with garlic, toasted sesame seed, soy sauce and vinegar). The stalk is good as a bamboo shoot texture alternative. Cut off any stalk that is woody and fibrous, into thin slices and blanch for longer than you would the florets, or use direct in soups. Also florets good lightly steamed, with a salad type dressing. Save the blanching liquid and use in soup, or for cooking cous cous or rice (pot liquor!).
I have never air fried broccoli. You might try that if you like the taste of coal.