Preheat oven to 350 degrees and position racks in the upper and lower thirds. Line 2 baking sheets with parchment paper.

To make cookies:

In a bowl, whisk together the flour, baking soda, ginger, cinnamon and salt. In the bowl of an electric mixer fitted with the paddle attachment, beat the butter and sugar at medium speed until fluffy, about 3 minutes. Beat in the egg and molasses. Add dry ingredients and beat at low speed until well incorporated, scraping down the sides. Drop tablespoon-sized dough onto the baking sheets, 3 inches apart. Bake the cookies for 10-15 minutes, until they have risen and fallen and are slightly firm; shift the sheets halfway through for even baking. Let cool slightly, then transfer the parchment paper to racks to cool completely.

To make the filling:

In the bowl of a electric mixer fitted with the paddle attachment, beat the butter with the confectioners’ sugar at medium speed until light and fluffy, about 3 minutes. Beat in the milk until well combined.

Arrange the cookies in pairs on a large working surface and spoon or pipe the filling onto the flat side of half of the cookie. Place the other cookie flat side down on top of the cookie with the filling and press down slightly until filling spreads to the edges.

Sandwich cookies can be stored in an airtight container between sheets of waxed paper for up to a week.