Preheat your oven to 350°F and grease a 9-inch square baking pan.
Whisk together flour, baking powder, baking soda, cinnamon, and salt in a bowl.
In a separate bowl, cream together the softened butter and light brown sugar until smooth.
Add the eggs one at a time, followed by the applesauce, mixing until well combined.
Gradually stir the dry ingredients into the wet until just incorporated.
Fold in the fresh chopped apples by hand.
Pour the batter into the prepared pan and bake for 30-35 minutes, or until a toothpick comes out clean.
Allow the cake to cool completely on a wire rack.
To make the frosting, melt butter and brown sugar in a saucepan over medium heat until bubbly.
Whisk in the heavy cream and cook for 2 minutes, then remove from heat and cool slightly.
Gradually whisk in the powdered sugar and sea salt until smooth and spreadable.
Spread the frosting evenly over the cooled cake before serving.