Preheat to 350°F (175°C). Grease and flour an 8×8 or 9-inch round pan.
Whisk together flour, ginger, cinnamon, baking soda, and salt.
In another bowl, whisk brown sugar, molasses, applesauce, oil, eggs, milk, and vanilla until smooth.
Stir dry into wet just until combined. Do not overmix.
Pour into pan. Bake 25–30 minutes until a toothpick comes out clean.
Cool in pan 10 minutes, then transfer to a wire rack.
Mix powdered sugar, cinnamon, molasses, and milk until smooth and spreadable.
Spread frosting evenly over cooled cake.
Slice and serve with tea, milk, or vanilla ice cream.