For the grilled artichokes: In a medium Dutch oven, combine the white wine, thyme, rosemary, bay leaf, 1/2 teaspoon salt and 1 cup water. Squeeze in the juice from the lemon halves and then drop the lemon halves into the pot. With a small spoon, scoop out the furry choke from each of the artichoke halves.
Place the artichokes in the pot, cut-side down. Cover the pot and bring the poaching liquid to a simmer over medium-high heat; reduce the heat to medium low to maintain a gentle simmer. Poach the artichokes for 10 minutes, or until the tip of a paring knife slides in easily at the thickest point. Remove the artichokes to a wire rack to drain and cool slightly.
For the bagna cauda: Place a small saucepan over medium heat. Add the olive oil, garlic and anchovy paste to the hot pan and stir with a wooden spoon, mashing the anchovy paste into the oil. Cook for about 3 minutes, or until the garlic is fragrant and just beginning to toast. Stir in the butter, one piece at a time, until completely incorporated. Season with a pinch of salt. Keep warm over low heat.
To finish the grilled artichokes: Preheat a grill pan over medium-high heat. Drizzle the artichokes with 1 tablespoon olive oil and sprinkle with 1/4 teaspoon salt.
Place the artichokes on the grill, cut-side down; grill until nicely browned, 4 minutes. Turn the artichokes and grill for another 2 minutes, pressing down gently for even browning.
Serve the artichokes with the bagna cauda on the side for dipping.