Season chicken evenly with Cajun seasoning, pressing it into the meat.
Heat 1 tbsp oil in a skillet over medium-high heat; cook chicken 4–6 minutes per side until blackened and cooked through. Rest 5 minutes, then slice.
Cook pasta per package; reserve 1/2 cup pasta water and drain.
In the same skillet, melt 3 tbsp butter with remaining oil; sauté garlic 30 seconds.
Add broth and cream; simmer 3–4 minutes until slightly thickened. Stir in Parmesan until smooth.
Toss pasta into sauce, adding reserved water to loosen if needed. Add sliced chicken on top and heat 1 minute. Season to taste.