Season both sides of each chicken breast with salt, pepper, and Cajun seasoning.
Melt the butter and olive oil in a skillet set over medium-high heat. Sear the chicken on both sides until golden brown, about 2 to 3 minutes per side. Remove the chicken from the pan and set aside.
To the same pan, add the remaining 2 tablespoons of butter, onion, and green bell pepper and sauté until softened, about 5 to 7 minutes. Stir in the garlic and additional Cajun seasoning, cooking until fragrant, about 1 minute.
Add the rice and chicken broth and bring to a simmer. Return the chicken to the pan, cover, and simmer for 15 to 20 minutes until the chicken is cooked through and the rice is tender and fluffy.