Preheat the oven to 400 degrees F. Drizzle a thin coating of olive oil on an 8x8 baking dish.
Gather a cutting board, sharp knife, kitchen scissors, lemon wedges, and spoon. Slice off about 1 inch of both the top and stem of each artichoke, and rub the cut surfaces with lemon juice. Trim the pointy outer leaves with kitchen scissors and peel off any rough leaves near the stem. Slice each artichoke in half lengthwise and rub the cut surfaces with lemon juice again. Scoop out the fuzzy choke with your spoon and discard. The artichokes may still brown a little even with the lemon juice, but that's OK because you are roasting them anyways.
Place the artichoke halves cut side up in the oiled baking dish. Stuff each half with a couple cloves of garlic and a few sprigs of herbs. Sprinkle with salt and pepper. Carefully flip each one over, making sure that the garlic stays in the center cavity. Cover the baking dish with foil and bake for 60 to 70 minutes, until the outer leaves are tender.
Let the artichokes cool slightly. To make a dipping sauce, melt 1/2 stick of butter in a bowl. Whisk in the juice of 1 lemon, and stir in some of the cooked garlic cloves. Taste and adjust seasonings as desired. To eat, remove a leaf, dip it in the melted butter sauce, and place it in between your teeth. Scrape the tender flesh from the leaf, discard what's left, and repeat. You can eat some of the smallest leaves towards the center of the artichoke, as well as the artichoke heart and stem, whole.