Add the artichoke bottoms to the boiling water, cook for 2 minutes (do not defrost first). Transfer them to a cutting board to cool, but let the water continue to boil. When the artichokes are cool, slice them in 1/4 inch slices. If the pieces are bigger than "bite-size," cut them in half.
While you are waiting for the water to boil, in a large sauté pan, heat the oil and butter over moderately high heat. Sauté the onion, salt, pepper, red pepper flakes, and zest, stirring occasionally, until the onion is translucent. This may take about 5-7 minutes. Add the sliced artichoke bottoms, and continue to stir, for about 4 minutes. Stir in the lemon juice and about 1/4 cup of the artichoke cooking water. Remove from heat.
While the onion and artichoke mixture is cooking, boil the pasta until al dente - about 2 minutes for fresh pasta, so wait until the artichokes are in the sauté pan, or 10 minutes for dried pasta. Make sure you reserve about 1 cup of the pasta cooking water, drain the pasta in a colander or the insert of the pasta pot. Add the pasta to the artichoke mixture. Add parsley and about 1/4 - 1/2 cup of the pasta water, so that the consistency is not dry but also not runny. Make sure everything is mixed really well. Then add 1 ounce of the cheese and toss well. Taste for salt and pepper. Serve with extra Parmigiano cheese on the side.
All you need with this pasta dish is a really great tossed green salad, a crusty baguette or French roll, and a great glass of white wine. (Be sure to choose a wine that can tolerate artichokes.) This dish can easily be served the next day at room temperature. Just be sure that you've thinned the pasta with enough pasta cooking water before putting it in the refrigerator so that it does not dry out overnight. This serves 5-6.