In a medium bowl, combine the peanut butter, sesame oil, lime juice, garlic, ginger, soy sauce, honey, water, and red pepper flakes. Whisk everything together until the mixture is smooth and well-blended.
In the meantime, warm 1 tablespoon of olive oil in a large nonstick skillet or wok over medium-high heat. Add the chicken and cook, browning on all sides until fully cooked, about 5 to 6 minutes. Transfer the chicken to a plate and set aside.
Add the remaining tablespoon of oil to the skillet. Toss in the broccoli and bell pepper, cooking for about 7 to 8 minutes until the vegetables are crisp-tender. Stir in the garlic, ginger, soy sauce, and half of the green onions, coating the veggies well, and cook for another minute.
Return the chicken to the skillet and pour in the sauce. Allow it to simmer for a few minutes until it slightly thickens. Sprinkle with the remaining green onions. Serve hot over rice, garnished with fresh cilantro and chopped peanuts.