In a medium saucepan over medium-high heat, saute broccoli and garlic in olive oil, stirring occasionally 7-10 minutes until broccoli is tender. Cover and set aside.
Make the alfredo sauce: Melt butter in a medium saucepan over medium heat. Whisk in flour until smooth. Very slowly add cream, whisking constantly until incorporated. Add milk to saucepan. Bring sauce to a low simmer over medium heat, stirring occasionally until sauce begins to thicken. Remove from heat and whisk in cream cheese, nutmeg, parsley, and parmesan cheese until smooth.
Place fettuccine in a large serving pot and pour sauce on top along with 1/2 cup reserved pasta cooking water. Add broccoli mixture and very gently toss noodles until evenly coated with sauce. Season pasta with salt and pepper to taste and serve immediately with extra grated parmesan cheese while warm.