Add the strands of lemon zest to the bagna cauda and stir to combine.
From each branch, remove most of the rosemary leaves, leaving 1 1/2 to 2 inches of 1 end covered in leaves. Place a bread slice on the work surface, and put a piece of mozzarella on it, followed by a fresh bay leaf, optional. Repeat 1 more time, ending with a slice of bread. You should have a stack that has 3 slices of bread and 2 slices of cheese. Continue with the remaining ingredients, to make 4 stacks total. (If you can't find large rosemary branches, substitute wooden skewers. Soak the wooden skewers in cold water for 20 minutes. As you're assembling, put a few sprigs of fresh rosemary throughout.)
Using the leafless end of a rosemary sprig, skewer 1 of the stacks through the center, leaving the rosemary leaves exposed on the other end. Turn the stack on its side and, entering from the opposite end, insert another rosemary sprig, leaving the leaves sticking out of the other end. Brush the entire skewer lightly with olive oil, and sprinkle it with a pinch of salt and pepper. Continue in the same manner with the remaining stacks.
To grill: Place the skewers on the hottest part of a charcoal or gas grill and cook for 1 to 2 minutes, turning frequently, until the cheese just begins to soften and the bread begins to brown.
To bake: Preheat the oven to 400 degrees. Place the skewers on a baking sheet and cook about 5 minutes.
Carefully transfer the skewers to a plate and drizzle with a generous amount of the bagna cauda. Scatter with parsley leaves.
Using a mortar and pestle, pulverize the garlic and anchovies into a smooth paste.
Transfer the paste to a small saucepan. Add the butter and olive oil and bring to a simmer over low heat. Continue simmering for about 5 minutes. Remove the pan from the heat and stir in the lemon zest and lemon juice. Season with salt, to taste.