Preheat the oven to 350°F (175°C). Line an 8×8-inch baking pan with parchment paper, leaving a 2-inch overhang on each side.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt until well combined.
In another medium bowl, beat the softened butter, brown sugar, and vanilla extract with an electric hand mixer on medium-high speed for 2 minutes until smooth and fluffy.
Add the eggs one at a time to the butter mixture, blending well after each addition. Scrape down the sides and bottom of the bowl as needed until the mixture is creamy.
Add half of the flour mixture to the wet ingredients and mix on low speed until just combined. Mix in the yogurt, then add the remaining flour mixture and fold until a few streaks of flour remain.
Toss 2 cups of chopped apples with the ground cinnamon. Fold the cinnamon-coated apples into the batter.
Spread the batter evenly into the prepared baking pan. Scatter the remaining 1 cup of apples over the top and drizzle about 2 tbsp of caramel sauce over them.
For the topping, blend the granulated sugar and ground cinnamon together and sprinkle over the top of the batter.
Bake for 35-45 minutes or until a skewer inserted into the center comes out clean.
Let the cake cool on a wire rack for at least 1 hour. Drizzle more caramel sauce over the top before serving.