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Heat oven to 200°F. In large pot, bring 4 quarts salted water to boil.
Add pasta and cook until al dente, about 9 minutes. Drain pasta, reserving cooking water. Set pasta aside.
Return cooking water to boil. Add broccoli; boil until just tender and bright green, about 2 minutes. Plunge immediately into bowl of ice water. Drain and set aside.
Meanwhile, sprinkle both sides of chicken generously with Cajun seasoning.
In large skillet, heat oil until shimmering. Arrange chicken in single layer and sauté, flipping once, until temperature on internal thermometer reaches 155°F, about 5 minutes per side. Remove from skillet, cover with foil and transfer to oven to keep warm.
Wipe skillet with paper towel and return to stove. Over medium-high heat, melt butter and heat until foaming. Stir in garlic; sauté until fragrant, about 30 seconds.
Whisk in wine; bring to simmer. Simmer until sauce has reduced by half, about 2 minutes.
Whisk in cream; simmer until slightly thickened, about 2 minutes. Fold in parmesan cheese. Season with salt and pepper.
Add reserved pasta to skillet and toss gently to coat. Arrange broccoli over pasta; top with chicken and garnish with additional parmesan cheese. Serve immediately.
