Boil the pasta: Bring a large pot of salted water to a boil. Add the pasta and cook until just shy of al dente, about 2 to 3 minutes less than the package instructions. Reserve 1½ cups of pasta water before draining.
Blanch the broccoli: Add the broccoli florets to the hot pasta water and let them sit for about 2 minutes, or until bright green and crisp-tender. Remove the broccoli with a slotted spoon, then drain the pasta.
Season the chicken: In a medium bowl, toss the chicken pieces with 1½ tablespoons olive oil, garlic powder, onion powder, salt, and black pepper until evenly coated.
Cook the chicken: Heat the remaining 1½ tablespoons olive oil in a large skillet over medium heat. Add the chicken and cook, stirring occasionally, until golden and fully cooked, about 6 to 8 minutes total. Transfer to a clean plate.
Build the sauce base: In the same skillet, melt the butter. Add the minced garlic and cook for about 30 seconds, just until fragrant. Pour in the reserved pasta water and scrape up any browned bits from the pan.
Add the pasta and cream: Add the drained pasta and heavy cream to the skillet. Toss gently and simmer for 2 to 3 minutes, or until the sauce begins to thicken slightly. Keep the heat low enough that the sauce does not boil.
Melt in the cheese: Add the Parmesan cheese and toss until fully melted and the sauce is smooth.
Finish the dish: Return the chicken and broccoli to the skillet and toss until everything is evenly coated in the sauce. Serve warm.