How to Make It
Squeeze the lemon halves into a large bowl of water. Working with 1 artichoke at a time, snap off the outer leaves. Using a sharp knife, cut off the top third of the artichoke. Peel the stem, leaving it long. Cut the artichoke in half lengthwise. Using a spoon, scrape out the furry choke. Cut each half lengthwise into thirds and add to the lemon water.
In a large saucepan, combine the 6 cups of water with the thyme, halved garlic, 2 tablespoons of the olive oil and the salt and bring to a boil. Add the artichokes, cover and simmer over low heat until tender, about 12 minutes. Drain and transfer to a large platter to cool.
In a blender, puree the chopped garlic with the anchovies, olives, vinegar and lemon juice. With the machine on, slowly pour in the remaining 1/2 cup olive oil and the grapeseed oil to make a smooth sauce. Scrape the dip into a bowl, stir in the lemon zest and season with pepper. Serve the artichokes with the anchoïade at room temperature.