This dish is 80% about the onions. Slice 4–5 large onions thinly. They will seem like too many but they shrink to almost nothing.
Melt butter with olive oil in a large, wide pan over medium-low heat. Add onions and salt. The salt draws out moisture and starts the caramelization.
Cook for 40–45 minutes, stirring every 5–7 minutes. Do not rush this by turning up the heat — you'll burn them. They go from pale → soft → golden → deep mahogany. Patience. Add a splash of water or broth if they start to stick.
Cook pasta while onions finish. Reserve 1 cup pasta water.
Once onions are deeply golden and jammy, add a splash of balsamic vinegar. Cook 1 more minute.
Toss with hot pasta, parmesan, and pasta water until creamy and combined.