Wash the Brussels Sprouts well. Trim the stem ends, removing any raggy outer leaves. Cut in half from stem to top and gently rub each half with olive oil. Heat 1 tablespoon of olive oil in a large skillet over medium heat; don't overheat the skillet! Place the brussels sprouts in the pan flat side down in a single layer. Sprinkle with a couple pinches of salt. Cover and cook for roughly 5 minutes; the bottoms of the sprouts should only show a hint of browning.
Once just tender, uncover, turn up the heat, and cook until the flat sides are deep brown and caramelized. Toss them once or twice to get some browning on the rounded side. Season with more salt, a few grinds of pepper, and a dusting of grated cheese.