Preheat the oven to 350 F.
Peel the apples about halfway down. Use a melon baller to scoop out the core of each apple (take care not to break through the bottom of the apples) and use the melon baller to scoop out extra apple flesh to make room for the custard, leaving a wall of apple about ¾-inch thick. Place the apples in a baking dish.
Whisk the sour cream, egg yolks, brown sugar, vanilla, cinnamon and nutmeg together and pour this into each apple, so that it comes up to the top but without spilling (you may have a little extra filling left, depending on how big your apples are). Pour boiling water around the apples to just come up a ½-inch and then loosely cover the dish with foil. You can insert toothpicks into the apples, if needed, in order to keep the foil from touching the custard.
Bake the apples for 25 minutes, then remove the foil and bake for about another 10 minutes, until the custard is just set. Cool the apples in the pan, then remove to a plate to chill until set, at least 3 hours. A little juice may come out of the apples as they chill.
To finish, place the apples on a cooling rack placed over a parchment-lined baking tray. Bring the granulated sugar, lemon juice and 3 Tbsp of water up to a boil over high heat. Continue to boil the sugar, without stirring and occasionally brushing the sides of the pot with water, until the sugar is an amber colour. Remove the pot from the heat and carefully spoon the hot sugar over the apples so that they are evenly and fully coated with a thin layer of the caramelized sugar. Let the sugar set for about 10 minutes, and then lift the apples to a plate to serve.
The apples should ideally be served after the sugar has set, but this step can be prepared up to 2 hours ahead. The apples themselves can be made a day ahead and stored refrigerated.