Set the oven to 350°F (around 177°C). Position the rack in the middle. Line a 9x13-inch pan with parchment paper, leaving an overhang.
Whisk flour, salt, baking soda, and spices in a large bowl. In a separate bowl, whisk molasses, melted butter, granulated sugar, and brown sugar until smooth. Whisk in egg and vanilla extract.
Mix wet ingredients with dry ingredients until just combined. Spread dough evenly in the prepared pan.
Bake for 25 minutes, or until the top is set and a toothpick inserted comes out with moist crumbs. Cool completely in the pan on a wire rack.
For the frosting, beat cream cheese and butter until smooth. Add confectioners' sugar, vanilla extract, allspice, cinnamon, and ginger. Beat on low for 30 seconds, then high for 2 minutes.
Spread frosting over cooled bars. Refrigerate for 30 minutes to set. Lift bars from the pan using parchment paper and cut them into squares.