1. Make the cookies: In a large bowl, whisk the flour with the cocoa powder, baking powder, baking soda and salt. In a standing electric mixer fitted with the paddle, cream the butter with the sugar until fluffy. Add the egg and vanilla and beat until blended. Beat in the dry ingredients at low speed. Scrape out the dough and pat it into two 6-inch disks. Wrap the dough in plastic and refrigerate for 30 minutes, or until chilled.
2. Preheat the oven to 350°. Let the dough soften at room temperature for 15 minutes. On a lightly floured surface, roll out 1 disk of dough to a scant 1/4-inch thickness. Using a floured 3-inch round or square cookie cutter, cut out as many cookies as possible and transfer them to 2 large ungreased baking sheets. Gather the scraps, press them together and chill briefly until firm enough to roll. Roll out the scraps and cut out more cookies.
3. Bake the cookies on the lower and middle racks of the oven for 12 minutes, or until they have puffed and the tops look dry. Let cool for 5 minutes on the sheets, then transfer the cookies to a wire rack to cool completely. Repeat with the second disk of cookie dough.
4. Make the filling: In a medium saucepan, combine the brown sugar with the cream and 4 tablespoons of the butter and bring to a boil. Cook over moderately high heat, stirring, until syrupy, about 2 minutes. Remove from the heat and let cool until thickened and barely warm to the touch, about 15 minutes. Transfer to a medium bowl and beat in the remaining 4 tablespoons of butter, the confectioners' sugar and the bourbon until the filling is fluffy.
5. Arrange half of the cookies on a work surface, tops down. Scoop 1 tablespoon of the butterscotch filling into the center and cover with the remaining cookies, tops up; press gently to form sandwiches. Transfer the cookies to a platter and serve.
Make Ahead: The sandwich cookies can be stored in an airtight container for up to 1 week.