Step 1
Make the cake: Position a rack in the center of the oven and preheat to 350°F. In a medium bowl, whisk together the flour, baking soda, cinnamon, ginger, and salt. To a small bowl, add the vinegar and ½ cup of hot water. In a stand mixer fitted with the paddle attachment, beat the sugar and butter on medium speed until a coarse paste forms, then beat in the sorghum syrup until well incorporated. Replace the paddle with the whisk attachment and, with the machine running, gradually add approximately half of the flour mixture, followed by approximately half of the vinegar mixture; then, gradually add the remaining flour mixture, followed by the remaining vinegar mixture, and continue whisking just until a smooth batter forms. Transfer the batter to a greased 9-inch cake pan and bake until the top is golden brown and a knife inserted into the center comes out clean, about 25 minutes. Set aside to cool to room temperature. Meanwhile, wash and dry the stand-mixer bowl and return it to the machine.
Step 2
Make the frosting: In the stand mixer fitted with the paddle attachment, combine the syrup and 1 tablespoon of room-temperature water until it resembles the consistency of maple syrup. Add the butter and vanilla extract and beat until smooth. Gradually add the powdered sugar, ¼ cup at a time, and beat until the mixture is fluffy and well combined.