Bring a large pot of water to a boil and cook 10 ounces bowtie pasta according to package directions. Drain and set the pasta aside.
Pat the 4 thinly sliced boneless skinless chicken breasts dry with paper towels and season both sides with 1/4 teaspoon salt and 1/4 teaspoon pepper.
Set up three shallow bowls for breading: Bowl 1 — mix 1/4 cup flour and 1/2 teaspoon Cajun seasoning. Bowl 2 — whisk 2 eggs. Bowl 3 — combine 3/4 cup panko breadcrumbs and 1 cup grated Parmesan cheese.
Dredge each chicken piece in the flour mixture, shake off excess, dip into the beaten eggs, then press into the panko-Parmesan mixture to coat evenly.
Heat 1/4 cup oil for frying in a large skillet over medium-high heat until shimmering. Add the breaded chicken and cook until golden brown and cooked through, about 5 minutes per side (internal temperature should reach 165°F/74°C). Transfer cooked chicken to a plate and keep warm or tent loosely with foil.
In the same skillet, add 1 tablespoon olive oil over medium-high heat. Add 1 red bell pepper (chopped), 1 yellow bell pepper (chopped), 1/4 cup diced onion, 1 cup sliced mushrooms, and 2 teaspoons minced garlic. Sauté until the vegetables are tender, about 4–6 minutes. Season with salt and pepper to taste, then transfer the vegetables to a bowl and set aside.
Wipe the skillet if there are large browned bits, then return it to medium heat and melt 2 tablespoons butter. Whisk in 1 tablespoon flour and cook for about 1 minute to form a roux.
Gradually whisk in 2 cups heavy cream and 3/4 cup low-sodium chicken broth. Bring to a gentle simmer, then add 1 cup Parmesan cheese and 1 tablespoon Cajun seasoning. Stir continuously until the cheese melts and the sauce thickens, about 3–5 minutes. Taste and adjust with salt and pepper as desired; add cayenne pepper to taste if you want more heat.
Add the drained pasta and the sautéed vegetables to the sauce and stir until everything is evenly coated and heated through.
Slice the fried chicken and arrange the slices on top of the Cajun pasta. Serve immediately.