In a small saucepan, simmer the apple cider over medium heat until it reduces to about ⅓ cup. Set aside to cool completely.
Preheat your oven to 350°F and line two large baking sheets with parchment paper.
Whisk together flour, baking powder, baking soda, cinnamon, ginger, and salt in a medium bowl.
In a large bowl, cream the butter and both sugars until light and fluffy (about 3 minutes).
Beat in the egg, vanilla, and the cooled apple cider reduction.
Gradually add the dry ingredients to the wet, mixing until just combined.
Use a cookie scoop to drop 2-tablespoon mounds onto the prepared sheets, spaced 2 inches apart.
Bake for 10–12 minutes until the tops spring back when lightly touched. Cool completely on a wire rack.
Make the filling: Beat cream cheese and butter until smooth. Add powdered sugar and vanilla, beating until stiff peaks form.
Pipe a ring of frosting on the flat side of half the cakes. Spoon a teaspoon of caramel into the center. Top with the remaining cakes and press gently.