Heat the butter and oil in a wide heavy pot over medium heat. Add the sliced onions and stir to coat. Cook uncovered for 40–50 minutes or longer, stirring every few minutes at first and more often as they darken, until soft and evenly golden brown. Reduce heat if they catch; add a tablespoon or two of water to loosen brown deposits, without leaving the onions swimming. Do not scrape black burned bits into the sauce.

Add the garlic and thyme and stir for about 1 minute. Pour in 1/2 cup of the measured broth and scrape up the golden-brown deposits from the bottom.

Add the remaining broth, dry rigatoni and pepper. Bring to a steady simmer. Cook uncovered, stirring frequently along the bottom, for about 12–16 minutes, or until the pasta is just tender. Start tasting near the package's suggested time. Add the reserved hot water a few tablespoons at a time if the liquid runs low before the pasta is ready.

When the pasta is tender, the sauce should be glossy and slightly loose. If very watery, simmer briefly, stirring; if thick, add a little hot water. Turn off the heat and wait about 1 minute.

Stir in the Gruyère in small handfuls until melted, then add Worcestershire and cider vinegar. Taste before adding any of the measured salt. Do not boil hard after adding the cheese; loosen with another splash of hot water if needed.

Divide among four bowls and serve promptly. Refrigerate leftovers within 2 hours in shallow containers, use within 3–4 days and reheat to 165°F (74°C), adding water and stirring to warm evenly.