First, quick soak the grits. Put the grits, water and salt in a big heavy pot. Bring to a boil, stir, turn off the heat, cover it and let it just sit there for 15 minutes.
While the grits are quick soaking, prepare the butter and cheese and leave out, so it is at room temperature when you start assembling things.
Meanwhile, roast the broccoli. Trim the florets off the stalks and cut into bite-sized pieces. Toss them with lots of olive oil and spread them out on a well-oiled baking sheet (we used a 1/2-sheet hotel tray). Crush a couple of big cloves of garlic, cut them into a few pieces each and scatter around on the backing sheet. Roast broccoli at 425ºF for about 20 – 25 minutes, or until nicely golden brown.
After the 15-minute quick soak, turn on the heat under the pot of grits, low. Add in the 1/2 cup cream. Bring to a boil, then reduce heat and simmer. Stir the grits often—I recommend using a wooden spoon—to keep them from sticking. If it looks like things are getting too dry or like they just want to stick to the bottom, add a little more water or cream or some milk.
Meanwhile, check the broccoli. It will be done when it is fork-tender and lightly browned.
The grits will be done when pretty much all the liquid has been absorbed and they are al dente—tender but still a little toothy. Add the butter and almost all of the cheese—save a little for garnish. Stir until everything is melted and well mixed.
Serve in soup plates or bowls, topped with the extra cheese and the broccoli.