Preheat the air fryer to 375° F. Toss the broccoli and onion with 2 tablespoons of olive oil, garlic powder, salt, pepper, and onion powder. Place the broccoli into the air fryer basket and roast for 10 minutes. Tossing halfway through. Reserve ½ cup of broccoli when finished.
Heat the remaining oil in a large saucepan over medium-high heat, add the garlic and cook for 1 minute, until fragrant. Add the broth and bring to a simmer. Add the broccoli and onions to the broth and simmer for 15 minutes.
Use an immersion blender to break down the broccoli until the soup is at a smooth consistency. Stir in the half & half and the cheese, cooking for 5 minutes more, stirring continuously, until the cheese is fully melted. Serve each bowl with the reserved broccoli on top.
Note: If you do not have an immersion blender, you can use a stand blender or food processor to puree the soup, then return to the pot and reheat before adding the half-and-half and cheese.