Mix the flour, baking powder, vanilla extract and cinnamon in a bowl.
2.
Break the eggs into a large bowl and beat them gently. Mix in the other ingredients one by one: the sugar, the yogurt, the apple balsamic vinegar, half of the flour mixture and half of the extra-virgin olive oil. Gradually continue adding the other half of the flour and oil. It should form a dense batter.
3.
Add the apples. Stir with a wooden spoon and pour the mixture into a greased cake pan.
4.
Bake the cake in a preheated oven at 356 ºF, for about 40 minutes.