Lightly coat an 8-inch square metal baking pan with cooking spray. Set prepared pan aside.
Weigh or spoon flour into a dry measuring cup; level with a knife. Combine flour and next 5 ingredients (through salt); stir with a whisk. Combine buttermilk and eggs. Combine 1/4 cup honey, 1/4 cup molasses, brown sugar, butter, and fresh ginger in a pan over medium heat. Stir constantly until sugar melts. Remove pan from heat; cool 5 minutes. Add egg mixture to honey mixture, stirring with a whisk. Add egg mixture to flour mixture; stir until well combined.
Pour batter into pan. Bake at 400° for 25 minutes. Place on a rack; pierce surface with a skewer. Combine 1 tablespoon honey, 1 tablespoon molasses, and 1 tablespoon water; brush over hot cake. Cool. Dust with powdered sugar.