Bring a large pot of salted water to a boil. Add the penne pasta and cook according to package instructions until al dente. Drain and set aside.
While the pasta cooks, heat olive oil in a large skillet over medium-high heat.
Toss the chicken pieces with Cajun seasoning until evenly coated.
Add the seasoned chicken to the skillet and cook for 5-7 minutes, stirring occasionally, until the chicken is cooked through and lightly browned. Remove the chicken from the skillet and set aside.
In the same skillet, add the sliced red, green, and yellow bell peppers. Sauté for 4-5 minutes until they begin to soften.
Add the minced garlic and crushed red pepper flakes to the skillet with the peppers and cook for 1 minute until fragrant.
Pour in the chicken broth and heavy cream, stirring to combine. Bring the mixture to a simmer and cook for 3-4 minutes until the sauce slightly thickens.
Return the cooked chicken to the skillet and stir to combine with the sauce and vegetables.
Add the drained pasta to the skillet and toss everything together until the pasta is evenly coated with the sauce.
Sprinkle the shredded cheddar cheese over the skillet and stir until the cheese melts into the sauce.
Season with salt and freshly ground black pepper to taste.
Remove from heat and garnish with chopped green onions before serving.