Start by preheating your oven to 350°F. Line an 8×8-inch baking pan with parchment paper, making sure to leave about a 2-inch overhang on each side.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt until it’s all evenly blended.
In another medium bowl, beat together the softened butter, brown sugar, and vanilla using an electric hand mixer on medium-high speed. Beat this mixture for about 2 minutes until it’s smooth and fluffy. Add the eggs one at a time, blending well after each addition and scraping down the sides and bottom of the bowl until it’s creamy. Mix in half of the flour mixture on low speed until combined. Then, add the yogurt. Add the remaining flour mixture and gently fold it in until just combined, leaving a few streaks of flour. Toss the chopped apples with cinnamon and fold 2 cups into the batter.
Spread this batter into your prepared pan and scatter the remaining 1 cup of apples on top. Drizzle around 2 tablespoons of caramel sauce over the apples.
For the topping, blend together the sugar and cinnamon and sprinkle it evenly over the top of the cake batter. Bake for 35-45 minutes until a skewer inserted into the center comes out clean. Let it cool on a wire rack for at least 1 hour, then drizzle more caramel sauce over the top before serving.
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