In a large saucepan, melt the butter on med-high heat. Add the garlic, sauteeing until fragrant. Mix in the flour until it gets pasty and immediately add in the broth and milk. Boil on high, stirring constantly, until the mixture starts to thicken. Reduce heat to low and stir in the salt, pepper, and 2 TB of parmesan. Add in roasted chicken and half the broccoli, coating everything in the sauce.