In a deep, medium-sized pot, add 3 parts water and 1 part extra virgin olive oil and bring to a soft boil. Clean each artichoke by removing the stem and external leaves. Place the cleaned artichokes in the pot with the boiling water and olive oil and cook for approximately 20 minutes. Once the artichokes are tender, drain the water and set the artichokes aside to cool.
Bring a large pot of salted water to a boil.
Wash all the mussels, scrubbing well to remove any grit or beard.
In a medium-sized sauce pan, pour ½ cup extra virgin olive oil and sauté the garlic cloves until they turn golden in color. Add the white wine, bring to a boil and simmer for 2 minutes until the liquid is reduced slightly. Add the cherry tomatoes, mussels and boiled artichokes. Cover the pot, and cook over brisk heat until all shells have opened (if any remain firmly shut after the rest have opened, discard them).
Meanwhile, drop the cavatelli into the boiling water, stirring to prevent the pasta from sticking. Cook until the pasta is al dente. Once the cavatelli is ready, drain the water from the pasta (reserving a cup of the pasta water) and add the pasta directly to the pan with the mussels. Toss the pasta, adding the salt, pepper and chopped parsley, to taste. If necessary, stir in the reserved pasta water to loosen the sauce. Serve on a warm plate.