Boil a large pot of salted water. Cook the spaghetti 1 minute less than the package instructions. Drain pasta, reserving 1 cup of the pasta cooking water.
Heat 3 tablespoons of olive oil in a large skillet over medium-high heat. Add the sliced Brussels sprouts and shallot/onion. Cook, stirring occasionally, until the vegetables start to brown around the edges, about 5 minutes.
Turn the heat down to medium and add the salt, pepper, garlic, and lemon zest. Cook until the garlic is fragrant, and the vegetables are tender, 2 to 3 minutes.
Stir in the lemon juice and ½ cup of the reserved pasta water. Add the spaghetti and drizzle with the remaining Tablespoon of olive oil. Toss the spaghetti with the cooked vegetables and continue to cook for a minute to allow the sauce to coat the pasta.
Turn off the heat, then stir in the grated parmesan. If the pasta sauce seems too dry, stir in a splash or two more of the pasta water. Taste and season with more salt and pepper as needed.
Serve the pasta on a platter or individual plates, topped with a drizzle of extra virgin olive oil, a few grinds of black pepper, and extra grated parmesan cheese.