For the artichokes: Trim the artichokes by removing the outer 2 layers of leaves and cutting 1/2 inch off the top. In a large Dutch oven, combine the water and lemons. Bring to a boil over medium-high heat. Add 3 artichokes and boil for 7 minutes. Using a slotted spoon, remove the artichokes and drain. Repeat with the remaining 3 artichokes. Cool the artichokes and cut in half lengthwise. Scrape and discard the fuzzy choke from each artichoke half. Set the artichoke halves aside.
For the dip: In a small skillet, cook the bacon until crisp, reserving 1 tablespoon bacon drippings. Crumble the bacon.
In a medium serving bowl, combine the crumbled bacon, bacon drippings, mayonnaise, sour cream, and rosemary. Chill thoroughly.
To serve: Spray a grill rack with nonstick, nonflammable cooking spray. Preheat the grill to medium-high heat. Arrange the artichokes, cut-side down, on the grill rack. Grill until tender, about 5 minutes on each side. Serve with the dip and garnish with a rosemary sprig.