Bring a large pot of salted water to a boil.
Add the pasta and cook until al dente according to package instructions.
Reserve 1 cup of pasta cooking water, then drain the pasta and set aside.
Heat a large skillet over medium-high heat.
Drizzle the chicken pieces with a little olive oil and season with salt and black pepper.
Add the chicken to the hot skillet and cook for several minutes until lightly golden and mostly cooked through.
Remove the chicken from the skillet and set aside.
In the same skillet, melt the butter.
Add the garlic and dried thyme, stirring for about 30 seconds until fragrant.
Pour in the heavy cream and lemon juice, then bring the mixture to a gentle simmer.
Return the chicken to the skillet and add the broccoli florets.
Partially cover and cook for 5–7 minutes, or until the chicken is fully cooked and the broccoli is tender.
Stir in the grated Parmesan cheese until melted and smooth.
Season with additional salt and black pepper as needed.
Add the cooked pasta to the skillet and toss everything together.
Add a little reserved pasta water if needed to loosen the sauce.
Continue tossing until the Creamy Broccoli Chicken Pasta is evenly coated in the silky sauce.
Serve immediately with extra Parmesan cheese if desired.