Step 1Heat the oil and 3 tablespoons of butter in a large, heavy skillet over medium-low heat. Add the onions, salt and pepper to taste, the thyme and bay leaf. Cook, stirring often, 10 minutes. Cover and continue cooking until the onions are very tender, about 15 minutes. Raise the heat to medium-high and cook the onions uncovered, stirring, until golden brown, about 5 minutes. Discard the bay leaf.
Step 2Bring a large pot of water to a boil and add salt. Add the pasta, stir and cook uncovered over high heat until tender but firm to the bite, about 7 minutes. Drain the pasta, reserving a few tablespoons of cooking liquid.
Step 3Add the pasta and remaining tablespoon of butter to the onions. Toss gently over low heat until the butter melts. Add 2 tablespoons of cheese and a dash of cayenne pepper and toss again. If the sauce is too thick, add 1 or 2 tablespoons of pasta cooking liquid. Add salt and pepper, if needed. Garnish with thyme sprigs and serve with extra cheese.