Line a 9-by-13-inch baking pan with an 18-inch sheet of plastic wrap. Spread the sherbet into the prepared pan; freeze until firm, about 1 hour.
Line a large baking sheet with paper towels. In a large bowl, whisk together the evaporated milk, eggs and vanilla. In a large skillet, melt 1 tablespoon butter over medium heat. Dip 4 bread slices in the egg mixture and cook until golden, 2 to 3 minutes per side; repeat with the remaining 3 tablespoons butter, bread and egg mixture. Drain on the prepared baking sheet; let cool. Discard the paper towels and line the pan with parchment paper, arranging the french toast on top. Freeze for 30 minutes.
Cut the sherbet into 8 equal slices. Top each of 8 chilled french toast slices with a slice of sherbet. Set the remaining french toast on top to form 8 sandwiches; freeze for 1 hour. Cut each sandwich diagonally before serving.