Place the shrimp, sliced green onion, egg white, diced pimentos, soy sauce, sesame oil, cornstarch, minced garlic, kosher salt, minced ginger, and cayenne pepper in a food processor and pulse 5 to 6 times until finely minced and combined. If you don't have a food processor, finely chop the shrimp and mix everything together in a bowl.
Pour about 1/4 inch of vegetable oil into a large nonstick skillet and heat over medium-high until hot but not smoking.
Cut each toasted bread slice in half diagonally and remove crusts if not already removed.
Spread an even layer of the shrimp mixture onto one side of each bread half, then sprinkle the tops with toasted sesame seeds.
Working in batches, carefully place the bread shrimp-side down in the hot oil and fry until golden, about 1 to 2 minutes. Gently flip and fry the other side until golden, about 1 minute.
Transfer the fried shrimp toasts to a paper towel–lined plate to drain briefly, then serve warm with sweet garlic sauce.
