Prepare pasta in salted water according to directions listed on package, add in broccoli during last 5 minutes of boiling pasta. Drain and set aside.
In a large deep skillet, melt butter over medium heat then add in garlic and flour and cook 1 minute, stirring constantly.
While whisking slowly pour milk into flour mixture, then cook, whisking to smooth lumps until mixture begins to thicken. Add in cream cheese and parmesan and whisk until melted.
Mix in heavy cream and season with salt and pepper. Stir in lemon zest and lemon juice.
Mix drained pasta and broccoli into sauce mixture along with the grilled chicken. Serve warm.