Steam broccoli florets until just tender, about 3 minutes, set aside.
Preheat broiler. Butter 13X9X2 inch broiler proof baking dish. Heat olive oil in heavy large pot over medium high heat. Working in batches, sauté chicken strips until just cooked through, about 4 minutes. Transfer chicken strips to bowl. Add sliced red pepper and garlic to pot and sauté until just tender, about 5 minutes. Add whipping cream, 1 cup grated Parmesan cheese and all the Monterey Jack cheese. Simmer until sauce thickens slightly, about 5 minutes. Add chicken strips and any juices in bowl to sauce. Add cooked pasta and cooked broccoli and toss until sauce coats the mixture, about 3 minutes. Season with salt and pepper. Transfer to baking dish. Sprinkle with remaining 1/2 Parmesan over. Broil until golden brown on top, about 3 minutes and serve.