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Brush the pizzelle cooking surfaces with some of the remaining melted butter. Spoon 1 tablespoon of batter in the center of the pizzelle iron. Close the iron and cook until deep golden, about 3 minutes. Transfer the pizzelles to a cooling rack. Cool completely. Trim off any excess pizzelle along the edges. Store the pizzelles airtight at room temperature.
To make the ice cream sandwiches: Gently spread the chocolate-hazelnut spread over 1 side of 12 pizzelles. Cut the ice cream carton to remove the ice cream from the carton in one piece. Starting at the widest end of the piece of ice cream, cut the ice cream crosswise into 1/2-inch-thick rounds. Place 1 slice of ice cream atop each coated pizzelle. (Alternately, the ice cream can be scooped atop the pizzelle.) Top each with a second pizzelle, pressing gently to adhere. Arrange the sandwiches on a baking sheet and freeze until firm. (Can be made 2 days ahead. Store airtight and keep frozen,)
Cook's Note: If the ice cream sandwiches are eaten immediately, the pizzelles will be crisp. However, if the sandwiches freeze overnight, the pizzelles will soften just enough to allow the sandwiches to be cut into wedges, if desired.