Make the pesto aioli: In a food processor, blend the garlic until roughly chopped. Add the basil, Parmesan cheese, lemon juice, salt and mayonnaise. Blend until a smooth, creamy sauce forms. (Store leftovers refrigerated for 2 weeks.)
Season the tomato slices with a pinch of salt. On each slice of bread, slather the pesto aioli. Build the sandwich by layering half the arugula and basil, the tomato slices, mozzarella slices, more arugula and basil, and then the aioli-slathered bread. Enjoy immediately, or wrap and refrigerate until serving.