Preheat the air fryer to 400° F. Toss the peppers with the oil, salt, and pepper. Place the peppers into the air fryer basket and roast for 15 minutes.
Meanwhile, heat the remaining oil and butter in a medium pot over medium-high heat. Add the onion, cooking for 7 minutes, until translucent, then stir in the garlic and cook for 1 minute, until fragrant. Add the peppers, basil, oregano, broth, and soy sauce, stirring together.
Bring to a boil, reduce to a simmer, cover, and simmer for 20 minutes. Use an immersion blender and blend until the soup is smooth. Stir in the half & half, cooking for 5 minutes, and serve.