Prep the onions and garlic: peel and cut the onions in half from root to tip, then thinly slice across the grain (about 1/8-inch). Mince the garlic. Uniform slices mean even caramelization; cutting across the grain shortens the fibers so the onions soften faster.

Bring a large pot of water to a rolling boil, add 1 Tbsp kosher salt (part of the 1 1/2 tsp total) and the pasta. Cook 1–2 minutes less than package directions to keep the pasta al dente — it will finish cooking in the sauce. Reserve 1–1 1/2 cups of the starchy pasta water before draining.

Heat a wide, heavy skillet over medium heat and add the olive oil. Once the oil shimmers (not smoking), add the sliced onions in an even layer. A wide pan gives a larger surface area so moisture evaporates faster and the onions brown rather than steam.

Cook the onions undisturbed for 6–8 minutes, then stir and continue to cook, stirring every 2–3 minutes, for a total of 25–30 minutes. Sprinkle the teaspoon of sugar over the onions after 8–10 minutes to accelerate and even out caramelization. Low-and-slow cooking converts onion sugars to flavorful brown, nutty compounds—this is the backbone of the dish.

When the onions are a deep amber (not burnt) and most moisture has evaporated, push them to the pan edge and add the butter. Let the butter melt to enrich the onions and build flavor. Stir in the minced garlic and red pepper flakes, cooking for 30–45 seconds until fragrant — briefly to avoid bitter burnt garlic.

Add about 1/2 cup of the reserved pasta water and the lemon juice to the pan and bring to a gentle simmer. The liquid will deglaze the pan, lifting up browned bits (fond) and concentrating flavor; the lemon balances the onions' sweetness and brightens the final sauce.

Reduce heat to low, add the drained pasta to the skillet, and toss vigorously (or use tongs) to combine. Add small splashes of reserved pasta water as needed to create a silky emulsion: the starch in the water binds fat (butter + olive oil) and cheese into a glossy sauce that clings to the pasta.

Remove the pan from heat and sprinkle in the grated Parmesan in two additions, tossing between each addition until melted and glossy. Salting should be adjusted now — Parmesan is salty, so taste before adding more salt. Adding cheese off the heat prevents the proteins from seizing and keeps the sauce smooth.

Season with black pepper, fold in chopped parsley, and finish with an extra drizzle (about 1 teaspoon) of good extra-virgin olive oil for gloss and aroma. Toss once more to marry flavors.

Serve immediately on warmed plates, garnishing with extra Parmesan and a small pinch of red pepper flakes if you like heat. Warm plates keep the pasta from congealing and maintain a restaurant-quality presentation.