Preheat the oven to 200 degrees F. Arrange the bread on a baking sheet and toast in the oven until slightly dry, about 10 minutes; let cool.
Meanwhile, drop the garlic into a food processor through the feed tube with the motor running. Add the ginger, 1 tablespoon at a time, until minced. Scrape down the food processor, then add the chicken, egg white, rice wine, cornstarch and 1 teaspoon salt; pulse into a smooth paste.
Spread 1 tablespoon of the chicken mixture on each piece of toast and sprinkle with the scallions. Put the sesame seeds in a shallow bowl and dip the chicken-covered side of the toasts in the seeds. (The toasts can be covered and refrigerated at this point for up to 3 hours.)
Heat about 1/2 inch of peanut oil in a deep skillet over medium-high heat until a deep-fry thermometer registers 325 degrees F. Working in batches, fry the toasts, chicken-side down, until golden brown and cooked through, about 4 minutes. Flip the toasts and cook 1 more minute. Drain on a paper-towel-lined plate. Season with salt and serve warm.