Preheat the oven to 375°F. Employing a twisting motion with a paring knife, remove the stem from each apple in a conical shape. Reserve the stems. Carefully scoop out the apple cores with a melon baller or small sturdy spoon, leaving ½ inch or so of flesh at the bottom of the apples.
Put the apples in an earthenware baking dish. Fill each apple with sugar, then add a splash of Cognac to each one. Replace the stems and sprinkle sugar generously over the tops.
Bake the apples on the top shelf of the oven for 45 minutes to an hour, until the skins burst. Serve at room temperature, making sure to spoon the juices over each portion.