Cinnamon-Roll Sugar Cookies

For the Cookies: Start with the basic sugar cookie dough. Shape the dough into 2 disks; roll each disk between 2 sheets of lightly floured parchment paper into a 7 x 10-inch rectangle, 1/4 inch thick. Chill until firm but pliable, about 10 minutes. Meanwhile, heat oven to 350 degrees F. Line 4 large baking sheets with parchment paper. In a small bowl, combine light brown sugar, granulated sugar and ground cinnamon. Sprinkle the disks with the sugar mixture. Starting with one of the long ends facing you, tightly roll up 1 disk using the parchment paper to neatly guide the dough (pressing the dough back together if it cracks). Working quickly, repeat with the remaining disk of dough. Return 1 roll of dough to the refrigerator. Cut the other roll into 1/4-inch-thick slices and place them on the prepared baking sheets, spacing them at least 1 inch apart. Bake, rotating the positions of the pans halfway through, until light golden brown, 12 to 14 minutes. Let cool on the sheets for 5 minutes before transferring to wire racks to cool completely. Repeat with the remaining dough, letting it soften just until pliable but still firm before slicing. For the Icing: In a small bowl, combine 1 cup confectioners' sugar with 4 teaspoon whole milk. Drizzle the icing over the cooled cookies.