Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper.
In a medium bowl, whisk together flour, ground ginger, baking soda, cinnamon, and cloves until well combined.
In a large bowl, cream the softened butter and brown sugar together until light and fluffy.
Beat in the egg, then stir in molasses and vanilla extract until combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Drop tablespoon-sized dough onto the prepared baking sheets, spacing them about 2 inches apart.
Bake for 12-15 minutes or until the edges are firm and the centers are set. Allow to cool on the baking sheets for 5 minutes, then transfer to wire racks to cool completely.
For the filling, beat the softened butter in a medium bowl until creamy.
Gradually add powdered sugar, mixing well after each addition. Add heavy cream and vanilla extract, beating until the frosting is fluffy. Add salt if desired.
Spread or pipe the vanilla cream filling onto the flat side of half the cookies, then top with remaining cookies to form sandwiches.
Serve immediately or refrigerate until ready to serve.
Notes
These cookies can be stored in an airtight container for up to one week. For best texture, let them come to room temperature before serving.