Combine all of the sauce ingredients in a food processor fitted with the S-blade or a high-powered blender. Process until all of the ingredients are combined, but the sauce still has visible chunks of peanut in it. Set aside.
Heat the olive oil in a large heavy-bottomed skillet over medium-high heat until shimmering. Add the onion, bell pepper, and mushroom to the pan; stir to combine and spread the filling out in an even layer so the veggies sauté instead of steam. Cook for 7-8 minutes, stirring once or twice and spreading the filling back out evenly each time. Add the minced ginger and hoisin sauce; stir to combine and cook for 1 minute longer. Stir in the water chestnuts, green onions, and raw walnuts to the pan and cook until the green onions begin to wilt, about 1 minute. Remove from the heat.
Assemble the Wraps
Layer lettuce leaves with sushi rice (if using), a heaping spoon full of filling and top with carrot ribbons, cilantro, and a sprinkle of sesame seeds.
If you are preparing the optional sushi rice for this meal 1 cup of uncooked rice will cook up to approximately 2 ½ cups of cooked rice. Since I cook gluten-free for my family, I used a homemade hoisin sauce; a store-bought sauce should work too, but I haven't tested it myself.