In a bowl, combine the chicken with the ginger, garlic, oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper.
Heat a slow cooker on high until hot. Add the chicken mixture to the slow cooker and brown, stirring occasionally, until golden, about 6 minutes. Reduce the heat to low and stir in the stock and soy and cook until the chicken is very tender, 6 to 8 hours.
Remove the chicken and shred with two forks, then return the chicken to the slow cooker. Increase the heat to high. Whisk together the eggs with the sesame oil and a pinch of salt. Whisk the egg mixture into the hot soup then season with salt and pepper to taste. Serve the soup topped with the scallion greens.
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