Grilled Cabbage Caesar

Name: Grilled Cabbage Caesar
Ingredients: 4 cloves garlic; 1/2 cup finely grated Parmesan (from about 2 ounces), plus more shaved Parmesan for serving; 1/4 cup plus 3 tablespoons olive oil, plus more for the grill grates; 1/4 cup fresh lemon juice; 3 tablespoons mayonnaise; 1 tablespoon Dijon mustard; 2 teaspoons anchovy paste, optional; 1 teaspoon white wine vinegar; 1 teaspoon Worcestershire sauce; Kosher salt and freshly ground black pepper; 1 medium head red cabbage (about 1 3/4 pounds); 4 ounces crusty sourdough bread, cut into 1-inch-thick slices