Remove the core from each apple without cutting all the way through, forming a hollow 2/3 inch to 1 inch in diameter. (If you are preparing the apples in advance, brush them with lemon juice to prevent discoloring.) Place the apples in a baking dish.
Fill the cavities with the honey and chopped ginger and top each one off with a small lump of butter.
Bake on the lowest oven rack for about 45 minutes, until the apples are tender. Check on the apples every once in a while—if there are signs that the honey is burning, turn down the heat.
Serve hot or warm, topping each apple with a teaspoon or more of the sour cream, if desired.