Put 1 clove garlic and the peeled ginger slice in a food processor and purée briefly.
Add the shrimp, drained water chestnuts, chopped green onion, 1 teaspoon of the egg white, chopped cilantro, and 2 tablespoons soy sauce to the processor.
Pulse the mixture in short bursts until the filling is chunky but holds together; do not purée into a paste.
Keep remaining wonton wrappers covered with a tea towel. Place a heaping teaspoon of filling in the center of each wrapper.
Brush the wrapper edges with the remaining egg white, fold to form a triangle or half-moon, squeeze out any air, then bring corners together and press to seal into a wonton.
Mix the sauce: combine 1 teaspoon finely minced garlic, 1 teaspoon sugar, 2 tablespoons soy sauce, 1 tablespoon roasted chili flakes in oil (drained), and 2 tablespoons black or rice vinegar in a small bowl; stir well.
Bring a large pot of water to a rolling boil and keep a cup of cold water nearby. Add a portion of wontons to the boiling water and stir gently so they don’t stick.
When the water returns to a boil, the wontons are cooked—use a slotted spoon to remove them and drain briefly on a plate; repeat, adding the reserved cold water between batches if needed to keep the pot boiling.
Divide the cooked wontons among small bowls, pour the spicy Sichuan sauce over them, top with sesame seeds and chopped green onions, and serve immediately.
