Preheat the broiler. Bring a large pot of water to a boil, salt it, add the pasta and cook until al dente (about 10 to 12 minutes, depending on the brand). When the pasta has been cooking for a couple of minutes, add the broccoli to the water and finish cooking everything together. Drain and return to the pot. Meanwhile, in a saucepan, melt the butter over medium heat. Add the garlic and cook for 2 to 3 minutes. Whisk in the flour and cook for 1 minute, then whisk in the milk and stock. Cook until the sauce thickens enough to coat the back of a spoon. Stir in 2 cups cheese and season with salt and pepper. Toss the pasta and broccoli with the sauce. Adjust the seasonings to taste. Place in a large casserole dish and top with the remaining 1/2 cup cheese and the chives or scallions, if using. Place under the broiler until bubbling and browned.