For the cucumber salad, mix everything together in a medium bowl. Let it sit for at least 10 minutes (longer is better - overnight is best!).
In a large skillet over medium-high heat, add the oil, cabbage, mushrooms, white/light green parts of the scallion, and salt. Stir to combine and cook for 6-8 minutes, or until the cabbage and mushrooms are softened and the water has cooked out.
Add the tofu, soy, sesame, ginger, and garlic and stir to combine everything and infuse the flavors together, about 1-2 minutes. If there is a lot of water in the pan from the tofu, simmer it so that it can evaporate. Stir in half of the green parts of the scallions (save the rest for topping).
For the dipping sauce, mix everything together in a small bowl.
Fill a bowl or plate wide enough to fit the rice papers with a thin layer of room temp water. Dip a rice sheet in it, making sure it is covered in the water, and let it sit for about 15 seconds. Remove, shake off excess water, and place on a cutting board. Repeat with another rice sheet and place on top of the first one (we want two layers!). You can use a paper towel to gently wipe off any excess water.
Use a sharp knife to cut the rice paper in half and then into quarters. Place about 1 scant tablespoon of the filling in the center of each triangle (it’s super important not to overfill and to keep the edges dry otherwise it won’t stick). With clean hands (if fingers are oily, the rice paper won’t stick well to itself), carefully fold two of the corners over and on top of each other (they should stick together) and then fold the other two sides over, stretching them so it’s nice and compact (see video!!). Transfer to a large plate or baking sheet lined with parchment. Repeat with remaining triangles. Then repeat with remaining rice sheets until the filling is used up.
Clean out the skillet used to make the filling and wipe dry. Over medium-high heat, add about 3 tablespoons oil (enough to generously coat the skillet). Once the oil is hot, add as many dumplings as you can fit without letting them touch, placing the folded side down, and let fry for about 4-5 minutes or until crispy and light golden. Flip and repeat on the other side. Transfer to a paper towel lined plate or baking sheet and let cook for 10 minutes before eating (they are very hot!).
Serve the dumplings over the cucumber salad with dipping sauce on the side. Top with green scallions and sesame seeds (if using).