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Preheat the oven to 425°. In a food processor, combine the scallions, mint, garlic, pepper, coriander and salt; pulse to mix. Add the shrimp and process to a thick paste. Add the eggs, sesame oil and chili-garlic sauce and process for 30 seconds. Transfer shrimp paste to a bowl.
Lightly butter 2 large baking sheets. Stack the bread slices and cut them into 3-inch squares, cutting off the crusts. Spread the bread with the shrimp paste and brush lightly with vegetable oil. Cut each square diagonally in half and arrange on the prepared baking sheets. Bake the shrimp toast in the middle and upper third of the oven for about 10 minutes, until the topping is cooked through; switch the pans halfway through baking.
Meanwhile, in a small bowl, whisk the fish sauce with the lime juice and sugar until the sugar dissolves.
Turn on the broiler. Working with one sheet at a time, broil the shrimp toasts 6 inches from the heat for about 2 minutes, until lightly browned and puffed. Transfer the toasts to a platter and serve with the dipping sauce.
The shrimp paste and the dipping sauce can be refrigerated overnight.