How to Make It
In a medium bowl, combine 1 cup of the oil, the parsley, savory, the chopped fresh thyme or 4 teaspoons dried, the rosemary, the lavender flowers, if using, and the pepper. Trim off the fat from the edge of the steaks. Put the steaks in a glass dish or stainless-steel pan and rub them on both sides with the marinade. Cover and marinate in the refrigerator overnight.
In a food processor, puree the butter with the olives, anchovies and 1/4 teaspoon of the salt. Cover and chill. Remove from the refrigerator 1 hour before serving.
Squeeze the juice from the lemon and add it, along with the lemon halves, to a large bowl filled with cold water. Working with one artichoke at a time, remove and discard the tough outer leaves, leaving the pale-green inner ones attached. Using a small knife, trim around the bottom of the artichoke below the leaves to get rid of the remaining bits of the removed leaves. Peel the stem, leaving it about 1 inch long. Cut off the top third of the leaves. Cut the artichokes lengthwise into 1/4-inch slices and put them in the lemon water to prevent darkening. Repeat with the remaining artichokes.
In a large frying pan, heat 1/3 cup of the oil over moderate heat. Add the garlic, the thyme sprigs or the remaining 1 teaspoon dried thyme, and the bay leaves and cook, stirring, for 30 seconds. Add the potatoes, 2 teaspoons of the salt, the wine and water. Cover, reduce the heat to moderately low and simmer until the potatoes are just tender, about 10 minutes. Drain the artichokes thoroughly and add them, with the remaining 1/3 cup oil, to the potatoes. Cover, bring back to a simmer and cook, stirring occasionally, until the artichokes are tender, about 10 minutes longer. If all the liquid evaporates, add up to 1/4 cup water during cooking. Set aside, covered.
Light the grill or heat the broiler. Sprinkle the steaks with the remaining 1/2 teaspoon salt. Grill or broil the steaks for 6 minutes. Turn them and cook to your taste, about 6 minutes longer for medium rare. Transfer the steaks to a carving board and leave to rest in a warm spot for about 5 minutes. Reheat the potatoes and artichokes while the steaks rest. Cut the steaks into thin slices.
To serve, put a mound of the potatoes and artichokes in the center of each plate. Arrange the steak slices, overlapping, on the vegetables and top with a spoonful of the room-temperature olive butter.
Notes
Substituting Large Artichokes: If you're using large artichokes, cut off the stems. Remove and discard all but the pale-green inner leaves and cut off the top third of the leaves. Cut the artichokes into quarters and remove the fuzzy chokes. Cut each quarter in half from top to bottom.