Open the cans of artichokes and drain the water well. Use your hands to squeeze our the majority of the brine, and place artichokes into a food processor. Pulse 8-10 quick times until the artichokes look roughly shredded (should be a tuna-like consistency). Do not over-blend.
In a medium bowl, add the shredded artichokes, vegan mayo, yellow mustard, diced onion, diced celery, diced pickle, tomatoes, and black pepper. Mix with a spoon until thoroughly combined.
Lightly toast bread. For each sandwich, lay a large piece of lettuce on the bottom slice of toasted bread and top with a generous layer of artichoke "tuna" salad. Top with the other piece of bread, cut in half using a sharp knife (if desired), and enjoy.