Sweet, deeply caramelized onions are simmered with garlic, beef stock, a splash of brandy, cream, and two cheeses to create a rich, velvety sauce that's tossed with perfectly cooked pasta.
Place a deep pan over medium heat and add the butter along with a drizzle of olive oil. Once the butter has melted, add the sliced onions and sauté for about 20-30 minutes, stirring occasionally, until the onions are deeply golden and caramelized. Reduce the heat if they begin to brown too quickly.
Stir in the garlic and thyme and cook for 30 seconds until fragrant. Sprinkle the flour evenly over the onions and cook for 1 minute, stirring constantly. Deglaze with the brandy, stirring and scraping up any browned bits, until most of the liquid (i.e. alcohol) has evaporated.
Increase the heat, then add the beef stock, Worcestershire sauce, and balsamic cream. Bring the mixture to a gentle boil.
Cook the linguine 2 minutes shy of al dente. Finish cooking it in the sauce for the final 1–2 minutes. Reserve about 1 cup (240 ml) of the pasta cooking water before draining.
Reduce the heat to medium and stir in the heavy cream. Once the sauce begins to thicken, remove from the heat and mix in the parmesan and graviera cheese. Season with salt and freshly ground black pepper to taste, stirring well until the cheese melts and the sauce is smooth.
Add the cooked linguine to the pan and toss thoroughly to coat the pasta in the sauce. If the sauce is too thick, add a splash of the reserved pasta water until it reaches your desired consistency. Serve immediately, garnished with freshly chopped basil.