Martha made this recipe on Cooking School episode 307.
Make a bowl of acidulated water: Fill a large bowl with cold water. Squeeze the juice of 2 lemons into bowl along with the lemon halves; set aside.
Using a serrated knife, cut off 1/2 inch from the top of an artichoke. Remove any small leaves from the bottom and trim stem so artichoke can stand upright. Use kitchen shears to trim sharp tips of artichoke leaves. Transfer to acidulated water, and repeat with remaining artichokes.
Fill a large pot with about 2 inches of water and add tarragon and a pinch of salt. Set steamer basket in pot, making sure water doesn't seep through holes. Bring to a boil, then reduce to a rapid simmer.
Place artichokes stem side up in steamer and season with salt. Cover pot and steam until bottoms of artichokes are very tender when pierced to the center with the tip of a paring knife, about 40 minutes. (Add more hot water if necessary to maintain level during cooking.)
To make the tarragon butter: Melt butter in a small saucepan over low heat, then stir in chopped tarragon and lemon juice, and season with salt.
Place an artichoke on each plate, remove the center leaves and with a spoon gently scrape down to the heart. Serve with lemon wedges and warm tarragon butter on the side for dipping.