In a large skillet, heat oil over medium. Add onions and thyme; season with salt and pepper. Cover, and cook, without stirring, until onions have released their liquid, about 5 minutes. Uncover, and cook, stirring occasionally, until deep golden brown, 25 to 30 minutes more. When bottom of skillet darkens, add a few tablespoons water, and scrape up browned bits with a wooden spoon (you may need to do this 2 or 3 times).
Meanwhile, bring a large pot of salted water to a boil. When onions have about 10 minutes left to cook, add pasta to water in pot, and cook until al dente. Reserve 1/2 cup pasta water; drain pasta, and return to pot.
Add onions and butter to pasta in pot; season with salt and pepper, and toss to combine. Gradually add enough pasta water to create a thin sauce that coats pasta. Serve pasta topped with Parmesan.
The secret to this pasta is cooking the onions until they're a deep golden brown; to keep them intact, stir only occasionally. Pappardelle, like other egg-based noodles, cook very quickly (even when sold dried). We like this pasta with butter-and-cheese sauces, such as this one.