Simmer 2 cups apple cider until reduced to approximately ¾ cup. Cool.
Reserve ½ cup reduced cider for the batter and ¼ cup for brushing.
Preheat oven to 350°F (175°C). Grease and parchment-line a 9 × 9-inch baking pan.
Peel, core, and finely dice the apple.
Whisk flour, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and cloves.
In another bowl, whisk melted butter, oil, brown sugar, and granulated sugar.
Whisk in eggs one at a time.
Add sour cream and vanilla.
Whisk in ½ cup cooled reduced cider.
Fold the dry ingredients into the wet mixture until nearly combined.
Fold in the diced apple just until no dry flour remains.
Transfer batter to the prepared pan.
Bake for approximately 35–42 minutes.
Cool for 10 minutes, then lightly brush the warm surface with reserved cider.
Cool the cake completely.
For the caramel, melt sugar in a light-colored saucepan until amber.
Carefully stir in butter, followed by heavy cream.
Remove from heat and stir in vanilla and salt.
Cool caramel for 10–15 minutes.
Drizzle over the cooled cake.
Slice into 12 portions and serve.