Preheat a grill or grill pan to medium-high heat.
Brush the chicken with 1 tablespoon olive oil and season both sides with 2 teaspoons of the Cajun seasoning.
Grill the chicken 8–10 minutes, turning once, until an instant-read thermometer reads 165°F (74°C); transfer to a plate, tent with foil and rest 5 minutes, then slice into strips.
Meanwhile, cook the pasta in a large pot of salted boiling water according to package directions; reserve 1/3 cup of the pasta cooking water, then drain.
In a medium saucepan, melt the butter over medium heat, add the minced garlic and flour, and cook 1 minute while whisking constantly.
Slowly whisk in the milk until smooth, then stir in 1–3 teaspoons Cajun seasoning to taste and bring the mixture to a boil while stirring constantly.
Reduce heat to low, stir in the Parmesan and the heavy cream (or half and half) until the cheese melts, then remove from heat and set the sauce aside.
Heat the remaining 1 tablespoon olive oil in a large skillet over medium-high heat. Add the sliced red and yellow peppers and the sliced red onion and sauté about 2 minutes.
Add 1 teaspoon Cajun seasoning and the sliced mushrooms to the skillet and sauté 2 more minutes, until vegetables are tender-crisp.
Toss the drained pasta with the sauce, adding reserved pasta water a tablespoon at a time to reach desired sauce consistency.
Add the sautéed vegetables and sliced grilled chicken to the pasta and toss to combine; adjust seasoning with additional Cajun seasoning or salt if desired.
Serve warm sprinkled with fresh parsley.