Prepare and steam the artichokes. Use kitchen scissors to snip off the tips of the leaves, and then cut off the stem and about ½ inch of the top with a sharp knife. Rinse well under cold water. Place the artichokes in a large pot filled with two inches water. Add the lemon slice and bay leaf. Bring to a boil over high heat, then reduce to a simmer and cover, continuing to boil until tender, 30-40 minutes. Remove from water and set aside to cool.
While the artichokes are steaming, make the rosemary olive oil and roast the garlic. To make the rosemary olive oil, pour the oil into a small saucepan set over medium heat. Add the rosemary sprigs. Bring just to a boil, then remove from heat. Let sit for 5 minutes. Remove the rosemary and discard it.
To roasted the garlic cloves, preheat the oven to 375º F. Place the unpeeled cloves in a shallow baking pan or pie dish and drizzle with 1 teaspoon of olive oil. Toss to coat and cover pan with foil. Bake for 20-25 minutes, until garlic is soft. Remove from oven and let cool enough to handle.
Make the dipping sauce. Squeeze the roasted garlic into the bowl of a food processor and discard the garlic skins. Add ½ cup rosemary olive oil, lemon juice, Dijon mustard, and salt. Blend until pureed, about 30 seconds. Pour into a small serving bowl and set aside.
To grill the artichokes, preheat grill to approximately 500ºF. Using a sharp knife, cut the artichokes in half. With a spoon, scrape out just the fuzzy, hairy center of the artichoke, leaving the leaves and heart. Brush well with some of the remaining rosemary olive oil. Place cut-side down on the grill and cook until the artichokes darken with grill lines, about 3 minutes. Use tongs to rotate a quarter turn; cook for another minute or two. Remove from heat and allow to cool for 2-3 minutes. Serve with dipping oil.
If you have leftover rosemary oil, it's perfect for dipping crusty bread or tossing with pasta and roasted vegetables.