Steaming is the the traditional way to cook one of these English cakes. The batter and jam are put into a heatproof bowl, then topped with parchment and a tea towel.
Butter a 5-cup heatproof pudding basin or mixing bowl. Cut a parchment circle to fit top of bowl; butter. Fill a large pot fitted with a steamer or pasta insert with enough water to come halfway up bowl (test this with an empty bowl); set aside.
Stir together flour, baking powder, and ground ginger in a small bowl. Put butter and sugar into the bowl of an electric mixer fitted with the paddle attachment; mix on medium speed until pale and fluffy. Mix in egg. Reduce speed to low. Add flour mixture in 3 batches. Add chopped ginger, honey, and milk; mix until just combined.
Spoon jam into bottom of prepared bowl. Pour batter on top; smooth with a spatula. Cover bowl with prepared parchment, buttered side down. Place a tea towel on top, and secure with kitchen twine. Knot opposing corners of towel together on top of bowl.
Bring water in prepared pot to a boil over medium-high heat. Carefully lower pudding bowl into steamer; cover pot. Reduce heat to medium-low. Steam 2 hours, checking occasionally to make sure water doesn't completely evaporate. Transfer to a wire rack; remove towel and parchment. Let cool 10 minutes. Run a small knife around edge of bowl; invert pudding onto a plate. Serve warm.