In a medium bowl, stir together the soy sauce, apple cider vinegar, 2 tablespoons olive oil, maple syrup, and liquid smoke. Add the eggplant slices and stir gently to coat. Marinate at room temperature for 30 minutes or more, stirring every 10 minutes or so to keep eggplant well coated.
Coat a large skillet with 2 teaspoons olive oil and place over medium-high heat. When hot, add eggplant slices in a single layer, working in batches if needed. Reserve marinade. Cook until the eggplant slices are browned on bottoms, about 3-5 minutes. Pour half of marinade into skillet (or less if you're working in batches) and cook another minute, until marinade cooks off.
Flip the slices and cook until they're browned on other side, about 3-5 minutes. Add the remaining marinade to the skillet and cook another minute, until the marinade cooks off again. The eggplant slices should be tender at this point.
Make the Sriracha cashew mayo:
Place the soaked cashews in a food processor fitted with the S-blade and process until a thick paste forms. Add lemon juice, milk, Sriracha, and salt. Continue blending until smooth and creamy, stopping to scrape down the sides of bowl as needed.
Assemble the sandwiches:
Slather one side of each bread slice with the mayo. Divide eggplant among four bread slices, followed by lettuce, then tomato slices. Top with the remaining bread slices, slice in half with a sharp bread knife if desired, and serve.
You will have more than enough Sriracha mayo for 4 sandwiches. However, I don't recommend scaling down the batch size unless you've got a small blending device that you're sure can adequately blend a small batch of cashews. You can freeze the extra mayo, use it some other recipes, or just store it in the fridge for a few days, as chances are you'll get a hankering for more of these sandwiches.