Bring a large pot of salted water to a boil. Cook pasta until al dente. Reserve ½ cup pasta water, then drain.
Season chicken with Cajun seasoning, salt, and pepper.
Heat olive oil and butter in a large skillet over medium-high heat. Cook chicken until golden and fully cooked. Remove and slice.
In the same skillet, melt butter over medium heat. Add garlic and sauté until fragrant.
Pour in heavy cream and simmer gently for 3–4 minutes.
Stir in Parmesan cheese until melted and smooth.
Add cooked pasta and chicken. Toss to coat, adding pasta water as needed.
Taste and adjust seasoning. Serve hot.