Preheat oven to 350°F. Line a 9" square baking pan with parchment paper and set aside.
In a small bowl, whisk the flour, ginger, baking soda, cinnamon, and salt together. Set aside.
Using a hand mixer or a stand mixer fitted with the paddle attachment, cream the butter with the brown sugar until smooth. Add the molasses, yogurt, and vanilla extract and beat to combine. Add the dry ingredients into the wet ingredients and mix on low speed until combined.
Pour the batter into the prepared pan and bake for 15-18 minutes, or until a toothpick inserted into the center comes out clean. Cool completely before frosting and slicing into bars.
To make the cream cheese frosting:
Using a hand mixer or a stand mixer fitted with the paddle attachment, beat the softened cream cheese and butter until smooth and creamy. Add the powdered sugar and vanilla extract and beat to combine.