Beat molasses, brown sugar, and butter together in a bowl until smooth. Mix flour, eggs, cinnamon, baking soda, ginger, and salt into molasses mixture until incorporated. Cover bowl with plastic wrap and refrigerate for 1 hour.
Preheat oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper. Heavily flour a cloth-covered work surface.
Roll dough on the floured cloth into 1/4-inch thick cookies; cut with cookie cutters if desired. Arrange cookies 1-inch apart on the prepared baking sheet.
Bake in the preheated oven until cookies are firm to the touch, about 8 minutes. Cool in the pans for 10 minutes before removing to cool completely on a wire rack.