Lay chicken breasts flat on a cutting board and slice horizontally into 2 thin pieces using a sharp knife.
Place 1/2 cup flour in a shallow bowl. Mix 1/4 cup Parmesan cheese and panko bread crumbs together in a separate shallow bowl. Beat eggs together in another shallow bowl.
Heat 2 tablespoons olive oil in an oven-proof skillet over medium heat.
Dredge chicken in flour, dip in cheese-bread crumb mixture, and coat with eggs. Place chicken in the hot skillet and cook until evenly browned, about 5 minutes per side. Transfer skillet to the oven.
Bake in the preheated oven until no longer pink in the center and the juices run clear, 8 to 10 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Bring a large pot of lightly salted water to a boil. Add penne and cook, stirring occasionally, until tender yet firm to the bite, about 11 minutes. Drain.
Heat remaining olive oil in a skillet over medium-high heat. Add mushrooms and onion; saute until tender, 3 to 5 minutes. Add yellow bell pepper, red bell pepper, white wine, and garlic. Cook until peppers are softened, 3 to 5 minutes more. Remove vegetables from pan.
Melt butter in the same skillet; add 2 tablespoons flour and whisk well. Add cream and chicken stock; whisk constantly until bubbling and thickened, 5 to 7 minutes. Remove from heat; stir in basil, Cajun seasoning, and red pepper flakes. Return pasta and vegetables to the sauce; mix in remaining 3/4 cup Parmesan cheese. Lay baked chicken on top.