Bring a large pot of salted water to boil. Cook penne pasta according to package directions until al dente. Drain and set aside without rinsing.
Season cubed chicken with Cajun seasoning, salt, and pepper. Heat 1 tablespoon olive oil in a large deep skillet over medium-high heat.
Add chicken in a single layer and cook without stirring for 2 minutes. Continue cooking, stirring occasionally, until golden and cooked through (5-7 minutes total). Transfer to a plate and tent with foil.
Add remaining oil and 2 tablespoons butter to the same pan. Sauté bell pepper and onion for 4 minutes until softening. Add jalapenos and garlic for the final minute.
Push vegetables aside and melt remaining 2 tablespoons butter in the center. Sprinkle flour over butter and whisk constantly for 2 minutes to create a roux.
Slowly pour in warm milk while whisking vigorously. Bring to a gentle simmer and cook for 3-4 minutes, stirring often, until thickened.
Reduce heat to low. Stir in heavy cream, parmesan, and cheddar cheeses until smooth and glossy. Season with salt and pepper to taste.
Return chicken and vegetables to the pan. Add cooked pasta and fold everything together until coated in the creamy sauce. Serve immediately garnished with cilantro and green onions.
