Preheat oven to 350 F/ 177 C. Butter and flour a 13 x 9 baking pan.
Pre-cook the apples: Peel, core, and dice apples. Melt 2 Tablespoons of butter in a skillet, add apples, 1/2 cup granulated sugar, cinnamon, and a few splashes of lemon juice. Cover and let simmer for about 10 minutes until apples are soft. Set aside.
While apples are cooking, mix topping ingredients together until crumbly, and set aside.
Prepare cake batter. Beat butter and sugar, then add eggs, yogurt, and vanilla. Mix well. Add dry ingredients, and mix until just moistened. Mix in cooked apples, then pour batter into prepared pan, and sprinkle with topping.
Bake at 350 F/177 C for about 40 minutes, until a toothpick inserted in the center comes out clean. Let cool for about 15 minutes before serving.