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Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.
In a large mixing bowl with an electric mixer at medium speed, beat together the butter, brown sugar and remaining 2 tablespoons granulated sugar until light and fluffy, about 3 minutes. Use a spatula to scrape down the sides of the bowl. Beat in the vanilla and egg until just combined, about 1 minute. Scrape down the sides of the bowl again, reduce the speed to low and gradually add the flour mixture, mixing just until incorporated, about 1 minute. The dough will hold together and be loose and sticky.
Divide the dough into 12 equal portions, about 1 heaping tablespoon each. Using your hands, roll each portion into a ball about the size of a golf ball. Roll the dough balls in the cinnamon-sugar mixture and place on the prepared baking sheets, spacing them 2 inches apart.
Bake until the cookies are set and the edges are slightly golden, 10 to 12 minutes. Let cool for 5 minutes, then transfer the cookies to wire racks to cool completely, about 30 minutes.
Meanwhile, divide the softened ice cream between 2 pint-size paper soup-to-go containers (which should be about the same diameter as the cookies), then freeze.
When ready to assemble the ice cream sandwiches, place the pecans on a small, flat dish. Using a serrated knife, cut 1-inch rounds of ice cream--straight through the containers--to make big beautiful circles of ice cream to fill the sandwiches. Add an ice cream round to the bottom of one of the cookies and then cover with a right-side up cookie, pressing down to form a sandwich. Repeat with the remaining cookies and ice cream rounds. Roll the sandwiches on end in the pecans, using your hands if necessary to help the nuts adhere to the exposed ice cream edge. Serve immediately or freeze, covered, for up to 3 days.