In a large pot of boiling, salted water, cook the pasta according to package directions. In the last 2 minutes or so of cooking, toss in fresh broccoli florets.
Meanwhile, in a medium saucepan, heat the olive oil over medium-low heat. Add garlic and cook, stirring frequently, until the garlic is golden, 1 to 2 minutes.
Whisk in the flour until smooth, about 2 minutes. Gradually whisk in the chicken broth, Greek yogurt, milk, nutmeg and Better Than Bouillon Roasted Chicken Base (if using, see notes above). Bring to a low boil, stirring constantly. Lower the heat and simmer, stirring gently, until the mixture thickens, about 3 minutes.
Drain pasta/broccoli, return to pan and add cooked chicken and sauce mixture. Gradually stir in 3/4 cup Parmesan, tossing to combine. Serve with more Parmesan, if desired.