Caramel Apple Cupcakes

Name: Caramel Apple Cupcakes
Ingredients: 1 stick salted butter, at room temperature; 3/4 c. granulated sugar; 2 large eggs; 1 tbsp. vanilla extract; 1 c. applesauce; 2 c. all-purpose flour; 1 1/2 tsp. baking powder; 1 tsp. kosher salt; 1/2 tsp. baking soda; 1/4 tsp. ground ginger; 1/4 tsp. grated nutmeg; 1 1/2 sticks (12 tablespoons) salted butter, at room temperature; 3 c. powdered sugar, sifted, plus more as needed; 3 tbsp. apple cider, plus more as needed; 1/4 c. caramel sauce, plus more for drizzling