Make the cookies: Position racks in the upper and lower thirds of the oven; preheat to 375 degrees F. Line 2 baking sheets with parchment paper; set aside. Whisk the flour, pumpkin pie spice, baking powder, baking soda and salt in a medium bowl. Beat the butter, granulated sugar, brown sugar and vanilla in a large bowl with a mixer on medium-high speed until pale and fluffy, about 5 minutes. Add the egg and beat until combined. Reduce the mixer speed to low and beat in the pumpkin puree, scraping the bowl as needed. Beat in the flour mixture until just combined, 30 seconds to 1 minute.
Scoop mounds of batter (about 1 1/2 tablespoons each) about 2 inches apart on the prepared baking sheets; shape into rounds with damp fingers. (Or you can transfer the batter to a piping bag fitted with a 1/2-inch round tip and pipe into 1 1/2-inch rounds.) Bake, switching the pans halfway through, until the cookies are golden and the tops spring back when gently pressed, 14 to 16 minutes. Let cool 5 minutes on the pans, then carefully transfer to racks to cool completely. Repeat with the remaining batter.
Meanwhile, make the filling: Stir the milk and instant coffee in a small bowl until the coffee dissolves. Beat the butter, confectioners' sugar and salt in a large bowl with a mixer on low speed until combined, then increase the mixer speed to medium high and beat until light and smooth, about 3 minutes. Reduce the mixer speed to medium low and beat in the milk mixture, scraping the bowl as needed, until smooth, about 1 minute. Refrigerate until slightly firm, about 20 minutes.
Gently sandwich the filling (about 1 tablespoon per sandwich) between the cookies. Roll the edges in sprinkles. If the filling is soft, refrigerate the whoopie pies about 30 minutes, then let sit at room temperature before serving.