For the Apples: Preheat oven to 375 degrees. In an oven proof skillet or non-stick pan, melt butter over low-medium heat and stir in sugar. Arrange apples, pretty side down (like a tart tatin) and distribute pecans on top and within the apples. Cook for 25 to 30 minutes until caramel is golden brown and apples are tender in the middle. Pour batter* on apples and carefully spread leaving a slight border of apples with no batter, (this will help in de-molding the cake). Place on a tray and bake in the oven for 25 to 30 minutes, until cake is golden-brown and delicious and a tester comes out with some crumbs stuck to it. Cool on a rack 15 minutes before de-molding on a large plate.
For the Batter: Mix the first 4 dry ingredients together. In a mixer with a paddle on high speed, cream the butter with the sugar until pale in color and fluffy. Add the vanilla, ginger and 1 egg and incorporate. Add the second egg and incorporate well. Add the yogurt, mix well, then turn mixer to slow and add the dry mix. Mix only until combined, do not over mix.
For the Star Anise Caramel Sauce: In a pan, gently heat the star anise with the calvados until a 50 percent reduction is achieved. Remove star anise. In a non-reactive, thick pan, mix sugar and water together and bring up to a simmer on medium heat. Cook until golden brown, then slowly add the infused calvados. Mix well then whisk in the cream. Be careful, mixture will boil rapidly. Reserve warm.
PLATING Slice the cake and place on a plate. Drizzle caramel sauce all over. May serve with some whipped cream if desired.