Prepare barbecue (medium heat). Sprinkle chicken with salt and pepper. Brush both sides with olive oil. Grill until cooked through, about 6 minutes per side. Dice chicken and set aside. Cook broccoli in medium pot of boiling salted water until crisp-tender, about 3 minutes. Set aside.
Melt butter in heavy large saucepan over medium heat. Add flour and cook 2 minutes, stirring constantly. Gradually mix in cream. Bring to boil, reduce heat, and simmer 10 minutes, stirring frequently. Add both cheeses and stir until sauce is smooth. Season to taste with salt and pepper. Add pasta, chicken, and broccoli to sauce; mix well. Garnish with chives and serve.