Preheat the oven to 350°F (180°C). Slice the top off a head of garlic, drizzle with olive oil, and wrap tightly in foil. Roast for 30 minutes, until the cloves are soft and golden.

Meanwhile, heat the butter and olive oil in a wide skillet over medium heat. Add the sliced onions and reduce the heat to low. Cook slowly, stirring occasionally, until the onions are deeply caramelized, golden amber in color, and jam-like in texture. This can take 30 to 40 minutes. If the onions begin to stick, add a splash of water and stir.

Once caramelized, add the thyme, roasted garlic (squeezed from the skins), and white wine. Let simmer for 1 to 2 minutes, just until the alcohol cooks off. Stir in the cream and milk, and simmer for 5 to 7 minutes, until the sauce thickens slightly. Season with salt and freshly ground black pepper to taste. Stir in the Parmesan.

Cook the linguine according to package directions until al dente. Drain and add the pasta directly into the sauce, tossing to coat evenly. Serve hot, topped with extra Parmesan.