Line four baking sheets with parchment paper or silicone baking mats. Set aside.
In a medium mixing bowl, whisk together flour, cinnamon, ginger, and salt. Set aside.
In a large mixing bowl, beat together butter and sugar for 1 to 2 minutes, until pale and fluffy. Add egg, and mix to combine. Scrape the sides of the bowl as needed.
Add half the flour mixture to the butter mixture, and mix until just combined. Add the rest of the flour and mix until completely incorporated.
Transfer the dough to a cookie press fitted with your design disc of choice. (Not all the dough will fit at once - you'll have to work in batches.). Press cookies onto the prepared baking sheets, about 2-inches apart. (I can fit 1 dozen per pan.)
Put the baking sheets into the refrigerator and chill cookie dough for 30 minutes.
Preheat oven to 350 degrees F.
Bake cookies for 12 to 14 minutes, or until the edges are still pale and cookies are set. Keep other baking sheets in the fridge until it's time to bake.
Remove cookies from the oven and let sit for 5 minutes on the pan. Then transfer to a wire rack to cool completely, about 30 minutes.
Once all the cookies are baked and cooled, make the buttercream. Add all the ingredients to a mixing bowl and beat until smooth.
Transfer buttercream to a piping bag (or ziploc bag with a corner cut off). Turn half of the cookies upside down. Pipe a dollop of frosting onto the bottom of each upside-down cookie. Place another cookie on top (bottoms facing each other) and press gently to spread the frosting to the edges.
Serve immediately or store in the fridge in an airtight container. I like to chill these cookies so the buttercream sets up and doesn't squish out the sides as much.
Notes
You will have leftover buttercream. You can make more spritz cookies and fill, frost some sugar cookies, or add a little on top of your french toast and let it melt!