Mix all ingredients together in a bowl and set aside.
Place 3 cups of water in a large mixing bowl. Peel and grate all the potatoes. Place grated potatoes in the bowl of water. Using your hands, wash well. Strain, squeezing out all the water and transfer to a dry mixing bowl. Mix in shallot, chickpea flour, apple cider vinegar, salt, and pepper. Mix well.
Heat oil a large non-stick pan over medium-high heat. Scoop out 1/4 cup of mixture and drop it carefully into the hot oil. Press it down a little bit with the back of a fork. Fry 3 to 4 pieces at a time. Flip latkes over when golden brown (about 5 minutes), and press down again to get an even thinner pancake. After 3 minutes lower the heat to medium and continue to cook for another 3 minutes. Remove from skillet and place on a tray lined with paper towels to drain. Transfer to serving plate.
Top each latke with 1 teaspoon vegan mayo, 1 tablespoon of tapenade, and a few slices of avocado.
Garnish with thinly sliced fresh basil and sprinkle with vegan parmesan cheese. Alternatively, dice the avocado and mix into the tapenade as the final layer.