Sauté onion in olive oil for 3-4 minutes, add garlic and cook for 1 minute. Stir in carrots, potatoes, chicken broth, tomatoes, thyme, rosemary, bay leaf, salt, and pepper.
Simmer for 20-25 minutes, add chicken and cream, heat for 5-10 minutes. Set aside.
Mix flour, salt, sugar, and baking powder. Add water and olive oil to form dough, knead for 3-4 minutes. Divide into balls, roll into 6-8 inch circles.
Place filling and mozzarella on dough, fold, and pinch edges to seal.
Preheat air fryer to 375°F (190°C). Brush calzones with butter, sprinkle Parmesan, and air fry for 10-12 minutes until golden.
Cool slightly, serve with marinara or enjoy as is. Garnish with parsley.