Add curry paste and coconut milk: Stir in the Thai red curry paste and cook for 1 minute, until fragrant. Slowly pour in the coconut milk, stirring constantly.
Simmer the curry: Bring the curry to a simmer and cook for 10 minutes, or until the sauce has thickened.
Add protein and vegetables: Add the chicken, red bell pepper, and broccoli to the pot. Cook until the chicken is cooked through and the vegetables are tender-crisp, about 5-7 minutes.
Season and serve: Stir in the cilantro, fish sauce, and lime juice. Season with salt and pepper to taste. Serve over cooked rice or noodles.
Optional additions: For a richer flavor, add a tablespoon of peanut butter or a handful of chopped peanuts. For a spicier curry, add a pinch of red pepper flakes.