For the artichokes, combine the water, thyme, garlic, salt, and bay leaf in a 3½ quart Dutch oven. Squeeze in the juice of half of a lemon and add the whole half to the pot as well.

Trim the stem of each artichoke almost all the way down to the bottom, then remove the outer layer of leaves and any additional leaves that are bruised or damaged. Cut 1½ inches off the top. Rub the cut side of the artichoke with the remaining half of the lemon. Using a pair of scissors, snip the thorns off of all of the remaining leaves.

Place each artichoke cut-side down, stemside up in the prepared pot. They should fist snuggly but have some space around each one. Place the pot over high heat and bring to a simmer. Reduce the heat to medium low, cover, and cook for about 20 minutes or until the tip of a pairing knife inserts easily into the base of the artichoke.

Remove the artichokes from the liquid and cool to room temperature. They can also be chilled completely in the refrigerator and served cold.

For the sauce: To a food processor add the parsley, basil, pine nuts, salt, and oil. Puree until smooth, scraping down the sides with a rubber spatula as needed. Add the cheese and pulse to combine. Serve alongside the artichokes, for dipping.