In a small bowl, whisk together 2 large eggs, 1/4 cup water, 1 Tablespoon sesame oil, 1/2 teaspoon kosher salt, and 1/8 teaspoon white pepper. Pour the mixture into a large plate or other shallow vessel (eg. a baking pan or pie plate) that is at least as big as a sheet of your rice paper.
Prepare a large baking sheet by lining it with parchment paper. Prepare a work station with the dish of egg mixture, a bowl containing 10 sheets rice paper, a bowl containing the minced green onions, and the prepared baking sheet.
Dip one sheet of the rice paper in the bowl containing the egg mixture, making sure it is coated on both sides (I like to kind of slide the rice paper through the egg mixture to do this). Hold the sheet up above the plate, letting any excess egg mixture drip back into the plate. Lay the sheet on one side of the prepared baking sheet. (Note that the rice paper will not soften immediately, but over time, the egg mixture will soak into the rice paper, allowing it to soften).
Sprinkle the rice paper with about 1 1/2 - 2 Tablespoons of minced green onion, avoiding the edges.
Repeat the above 2 steps with 4 more sheets of rice paper and approximately half of the green onions, leaving the top sheet of rice paper (the 5th one) without green onions. Repeat this whole process again with the remaining 5 sheets of rice paper and the second half of the green onions to make a second pancake. Make sure that all of the layers of rice paper on both pancakes are completely softened before you start cooking.
Heat 1 Tablespoon of canola oil in a large non-stick skillet over medium-high heat. When the skillet is hot, add one of the green onion cakes and cook until the bottom is well browned (about 5 minutes). Flip the pancake, and cook until the second side is well browned, about 3-4 more minutes. Reduce the temperature to medium if the pancake is browning too quickly. Repeat with the second pancake, adding up to 1 Tablespoon of additional oil if needed.
Cut each pancake into 6 wedges and serve hot. Serve with sriracha, Thai sweet chili sauce, or chili garlic sauce if desired.
Make Ahead Instructions
Note that the recipe can be completed up to the end of step 5 in advance. At this point, the baking sheet containing the pancakes can be lightly covered with a clean dish towel or plastic wrap and placed in the fridge for up to 4 hours. Follow the remaining steps to cook the pancakes.
Notes
The size of the rice paper I used was 22cm (~9"). If your rice paper is significantly smaller than this, I would recommend making 3-4 pancakes instead of just 2.