Preheat oven to 325 degrees F. Spray an 8-inch baking pan with nonstick cooking spray, then line with parchment paper, allowing the excess to hang over the edges of the pan; set aside.
In a small saucepan, bring the dried apples and apple cider to a simmer over medium heat. Continue to cook until the liquid evaporates and the mixture appears dry, about 15 minutes. Remove from heat and set aside to cool to room temperature.
In a small bowl, whisk together the flour and baking soda; set aside.
In another small bowl, whisk together the sugar, cinnamon, nutmeg and cloves.
Measure 2 tablespoons of the sugar-spice mixture into another small bowl and stir in the brown sugar, pecans and oats. Add the softened butter and rub the mixture between fingers until the butter has been fully incorporated; set aside.
In a food processor, process the cooled dried apple mixture and applesauce until smooth, 20 to 30 seconds, scraping the sides of the bowl as needed; set aside.
In a large bowl, whisk together the egg and salt. Add the sugar-spice mixture and whisk continuously until well-combined and light colored, about 20 seconds. Add the melted butter in three additions, whisking to combine after each. Add the applesauce mixture and vanilla extract, and whisk to combine. Add the flour mixture and, using a rubber spatula, gently fold until just combined and evenly moistened.
Transfer the batter into the prepared pan, smoothing the top into an even layer. Pinch the reserved streusel mixture into hazelnut-sized clumps and sprinkle evenly over the batter. Bake until a thin knife inserted into the center of the cake comes out clean, 35 to 40 minutes. Cool on a wire rack to room temperature, about 2 hours. Remove the cake from the pan by lifting the parchment overhang and transfer to a cutting board. Cut into squares and serve. The cake can be wrapped tightly in plastic wrap and kept at room temperature for up to 3 days.
Notes
Notes:
This recipe can be doubled and baked in a 9x13-inch pan. The dried apple and apple cider mixture will need about 20 minutes to cook, and the baking time should be increased to 45 minutes.
This is a very moist cake, so it is better to err on the side of overdone when checking for doneness.
While apple cider is preferred, apple juice can be substituted.