Cook the pasta and broccoli: Bring a large pot of salted water to a boil. Add the pasta and cook according to the package directions, adding the broccoli florets during the last 2 minutes. Reserve 1 cup of the pasta water, then drain.
Make the roux: In a large skillet over medium heat, melt the butter. Add the garlic and cook for 1 minute until fragrant, then sprinkle in the flour and stir constantly for 1 minute.
Build the sauce: Slowly whisk in the chicken broth, then the heavy cream. Bring to a simmer and cook for 3-4 minutes, stirring often, until it starts to thicken.
Add the chicken: Stir in the shredded chicken, salt, and pepper, and cook for 2 minutes until warmed through.
Finish with parmesan: Remove from heat and stir in the parmesan until melted and smooth.
Combine: Add the drained pasta and broccoli to the skillet and toss to coat. If the sauce is too thick, stir in a splash of the reserved pasta water.
Serve: Top with red pepper flakes if using and serve warm.
Notes
Blanch the broccoli in the pasta water during the last 2 minutes instead of simmering it in the sauce, it keeps the color bright and the texture crisp-tender. If cooking chicken fresh instead of using rotisserie, cook it to 165°F before shredding.