Bring a large pot of salted water to a boil. Add the pasta and cook according to the package instructions until al dente. Add the broccoli florets to the same pot for the last 2-3 minutes to steam them. Reserve 1-2 cups of pasta water before draining.

Meanwhile, heat the olive oil and butter in a large, deep pan over medium heat. Once the butter has melted and starts to foam, add the chicken. Season with salt and black pepper, and cook for 5-7 minutes or until the chicken is cooked through.

In the same pan, add the chopped onion and cook for 3-4 minutes until softened and translucent. Add the chopped garlic and cook for an additional 1-2 minutes, stirring frequently to prevent burning.

Pour the dry white wine into the skillet, scraping up any browned bits from the bottom of the pan. Simmer until the wine has mostly reduced.

Stir in the fresh thyme and red chili flakes, then add the half and half (single cream). Bring the mixture to a gentle simmer, then reduce the heat and cook for 5 minutes, allowing the sauce to thicken slightly.

Add the cooked pasta and broccoli to the skillet, tossing gently to mix everything together. Use a bit of the reserved pasta water to loosen the sauce if it looks too thick.

Stir in the grated Parmesan cheese and season with additional salt and black pepper to taste.

Serve immediately, garnished with extra Parmesan cheese and a sprinkle of fresh thyme if you like.