Mix all the dough ingredients to make the sugar dough, using a stand mixer with a flat beater.
Once the dough starts to shape together around the flat beater, remove it from the stand mixer and shape it into a ball. You don't want the dough to become too elastic, so don't overwork the dough. Flatten the dough into a disc in order for it to get colder faster. Place the dough disc in a plastic storage bag and let it rest in the refrigerator for 1 hour.
Meanwhile, make the caramel. Melt the butter, and add the sugar. Stir a bit, then add the rosemary. Allow the caramel to get thicker but not too brown, and add a few drops of lemon juice.
Pour the caramel into your round baking dish or four individual ones.
For the apples, peel and rub them with some lemon juice to avoid oxidation.
Cut them into big chunks and place them in the baking dish. Be generous with the apples; it doesn't need to be perfect.
Take out the dough from the fridge. Dust flour on your surface, and then dust your dough. (in French "fleurer").
Roll out the dough and shape it (cut) to the size of your baking dish. Cover the apples with the rolled-out dough and close the dough on the baking dish. Make small holes with the point of a knife.
Bake the apple tatin in the oven for around 35 to 40 minutes at 375 °F.
Turning out the tatin from a hot baking dish can be a little tricky; you need to wear oven gloves.
Serve the warm Tarte tatin with one generous scoop of vanilla ice cream.