Place a pot of water over high heat and bring to a boil. While you are waiting for the water to boil, cook the brussels sprouts.
Set a large skillet over high heat. Add the oil, and wait until it begins to lightly smoke. Add the brussels sprouts in a single layer (if you have to do it in batches, do). Leave them for a minute or two, and don’t fuss with them too much—you want them to develop a nice golden color but not get soft. Toss, and continue cooking until the sprouts are just cooked through (imagine al dente pasta).
Remove the brussels sprouts from the pan and put them on a baking sheet. (This way, they won’t continue to cook in the pan.) Season them with salt and freshly ground black pepper, and the juice of a lemon.
Add the pasta to the boiling water. While the pasta is cooking, set a large skillet over medium heat, and add 1 tablespoon of butter and the
garlic. Cook the garlic until it is soft and just beginning to brown
around the edges. When the pasta is one minute away from being cooked al dente, transfer it to the skillet with the garlic.
Add a little reserved pasta water and the stock to the skillet along with the rest of the butter and stir. When the stock has reduced, add the cheese and brussels sprouts. When the cheese has melted, turn off the heat. Add the egg to the skillet and stir vigorously. You don't want the egg to scramble—you want it to add a creamy richness to the sauce. Taste the pasta and add
more salt if necessary.
Serve the pasta with an extra sprinkle of grated pecorino on top and a
few cracks of freshly ground black pepper.