Boil the pasta until it is al dente (it will cook a little bit more in the cheese sauce, so don't overcook it), adding the chopped broccoli into the boiling water a minute or two before the pasta is done cooking. Drain, season lightly with salt and set aside.
In a medium saucepan over medium heat, melt the butter. Then whisk in the flour to make a roux.
Cook the roux (it should be bubbling, but not turning brown) for a couple of minutes, and then slowly add the milk, a little at a time, whisking well between each addition to keep the sauce smooth.
When all the milk has been added, gently bring the mixture up to a simmer.
Once you start to see the mixture simmer, take it off the heat, and add the shredded cheese, starting with just 1/4 cup or so, and mixing it in to the sauce. Then add the the rest of the cheese in a couple batches, mixing each addition in.
When the cheese has melted into the sauce, toss it with the pasta and broccoli, and transfer everything to an oven safe baker.
Top with the 1/2 cup extra shredded cheddar cheese, and bake for about 15-20 minutes until bubbling. You can broil the baked pasta for another couple of minutes for a browned top if you like.