Place garlic into the bowl of a mortar or a small bowl. Add salt; mash until garlic is a smooth paste. Add lemon juice, lemon zest, cayenne, and mayonnaise. Stir until well blended.
Toast bread slices. Spread generously with the lemon aioli. Place anchovies, skin side up, on top of the aioli, about 4 fillets per tartine. For easier serving, cut the tartines in half at this point; set them back together before topping with the veggies.
Place carrot strips, endive, and radish slices in a bowl. Sprinkle with a splash of fresh lemon juice and a drizzle of olive oil. Toss lightly until veggies are dressed.
Place endive on the anchovies; arrange the carrots and radishes on the tartine. Top with edible flower petals and baby parsley.
