Set a large pot of water over high heat on the stove. Cook and drain the pasta, according to the directions on the package.
While the pasta is cooking, melt 2 tablespoons of butter in a large, deep skillet over medium heat. Add the garlic and the broccoli, toss to coat. Then, add 2 tablespoons of water, and cover with a lid.
Cook for 2-3 minutes until bright green and tender-crisp, stirring occasionally. Then, add the chicken, toss to combine, and transfer everything to a large bowl next to the stove.
Melt the remaining 2 tablespoons of butter in the same skillet over medium heat. Then, whisk the flour into the melted butter to create a roux.
Slowly add the chicken broth, whisking to combine. Then, increase the heat to medium-high and bring to a simmer. Simmer for two minutes, then add the cream.
As soon as it simmers again, reduce heat to low. Let it simmer on low, stirring constantly for 8-10 minutes. The sauce will thicken slightly as it cooks.
Add the cooked pasta to the pan with the sauce and toss to coat. Then, return the chicken and broccoli and toss gently to mix throughout. Season to taste with salt and pepper. Then, sprinkle with Parmesan cheese and parsley before serving.