Thinly slice the onions using a mandoline or a food processor's slicing disk for evenness.
Place the sliced onions in a large, dry skillet over medium-low heat and cover.
Stir every 5 minutes. The onions will first release a lot of water, then begin to dry out.
After 20-30 minutes, when the onions begin to brown and stick to the pan, remove the cover.
Add 1/3 cup (720 ml) of the olive oil, a generous pinch of salt, and a few grinds of pepper. Increase the heat to medium and continue to cook, stirring frequently, until the onions are uniformly deep brown, soft, and almost jam-like, about 10-20 minutes more.
While the onions are finishing, bring a large pot of salted water to a boil.
Cook the pasta according to package directions until al dente (tender but not mushy).
Before draining, reserve about 1 cup (240 ml) of the starchy cooking water.
Drain the pasta and transfer it to a large, warm serving bowl.
Add the caramelized onions, the remaining 2 tablespoons (30 ml / 1 fl oz) of olive oil, and a generous amount of Parmesan cheese.
Toss everything together, adding a splash of the reserved pasta water as needed to create a silky sauce that coats the pasta.
Taste and adjust seasoning with more salt and pepper if needed.
Serve immediately, passing extra Parmesan at the table.
Don't rush the caramelization process. Low and slow heat is the key to developing the onions' deep, sweet flavor without burning them.