Preheat oven to 350F. Butter 9 inch diameter cake pan and coat with coarse sugar.
Peel and core apples and thinly slice.
Melt 1/4 cup butter in a large heavy skillet over medium heat. Stir in 3/4 cup sugar, water and cinnamon and bring to a boil. Add apple slices and cook till just tender, turning frequently, about 10 minutes.
Remove apples from pan using a slotted spoon and arrange at the bottom of the cake pan.
Continue cooking liquid until it is thick and syrupy, resembling caramel, about 5 minutes. Pour over apples.
Combine flour, baking powder and salt into a small bowl. Whisk remainly 1 cup sugar, egg yolks, eggs, and vanilla in a large bowl to blend. gently stir in dry ingredients. Fold in 1/2 cup cooled melted butter.
Pour batter over apples in pan and bake until toothpick comes out clean, about 45 minutes. Cool cake in pan for 5 minutes. Run sharp knife around the side to loosen and turn cake out onto a platter.
