Bring ginger, honey, and water to a boil over high heat in a medium pot. Reduce heat to medium and let simmer 5 minutes.
Meanwhile, peel pears and halve lengthwise. Use a melon baller to scoop out cores. Gently lower pears into pot. Cover surface with a round of parchment paper.
Cook pears until a paring knife slides easily into them, about 10 to 15 minutes, depending on ripeness. Remove with a wooden spoon and transfer to a bowl. Pour poaching liquid over. Serve warm or chilled.
To serve, place pear halves in a bowl and drizzle with melted chocolate.