1. Slice the onions into thin half-moons.

2. Warm 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium heat. Add the onions and cook for about 10 minutes, stirring often, until softened.

3. Stir in the salt, reduce the heat to low, and cook for 30 to 40 minutes, stirring frequently, until the onions are deeply golden, sweet, and jammy. Add a splash of water as needed if the pan gets dry.

4. Cook the pasta in salted boiling water until just shy of tender. Reserve 1 cup of pasta water, then drain.

5. In the same skillet, warm the remaining 1 tablespoon olive oil and 1 tablespoon butter over medium heat. Add the garlic and red pepper flakes, if using, and cook for about 1 minute.

6. Pour in the heavy cream and bring it to a gentle simmer. Stir in the thyme.

7. Add the Parmesan a little at a time, stirring until the sauce is smooth and creamy.

8. Add the pasta, caramelized onions, lemon juice, and 1/4 cup reserved pasta water. Toss well over low heat for 1 to 2 minutes, adding more pasta water as needed until glossy and saucy.

9. Season with black pepper and more salt if needed. Serve right away with extra Parmesan and lemon zest, if desired.