Preheat oven to 325 degrees F. Butter a 9-inch square metal baking pan and line with parchment, leaving a 2-inch overhang on 2 sides.
Whisk together flour, baking powder, cinnamon, and salt.
In a large bowl, with a mixer, beat butter and 1 cup brown sugar until fluffy. Add vanilla and eggs; beat until combined.
Reduce speed to low; add flour mixture in 3 additions, alternating with 2 additions applesauce, and beat until combined.
Transfer batter to pan and smooth top. Arrange apples in 3 rows on top, overlapping slightly.
Bake until a toothpick inserted in center comes out with a few moist crumbs attached, 1 hour 15 minutes. Transfer pan to a wire rack and let cool 20 minutes.
While the cake bakes, cook the cider and remaining 1/4 cup brown sugar in a medium saucepan over medium-high until thick and syrupy, 15 to 18 minutes.
Using parchment, transfer the cake from pan to a serving plate and then remove/discard the parchment. Brush glaze over top and sides and let sit until cool and set, 1 hour.