Preheat the oven to 350°F. Grease a 9 x 13-inch baking pan or line it with parchment paper.
In a large mixing bowl, whisk together the flour, granulated sugar, brown sugar, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt. Break up any brown sugar or leavener lumps.
Add the applesauce, dairy-free milk, oil, apple cider vinegar or lemon juice, and vanilla.
Fold everything together gently just until no dry pockets of flour remain. Do not keep mixing once the ingredients are combined.
Transfer the batter to the prepared pan and spread it evenly into the corners.
Bake for 32 to 38 minutes, checking around the 32-minute mark. The cake is done when the center is set, the top springs back lightly, and a toothpick inserted near the middle comes out clean or with a few moist crumbs.
Place the pan on a wire rack and let the cake cool completely before frosting.
While the cake cools, melt the vegan butter for the apple topping in a skillet over medium heat.
Add the diced apples, maple syrup, cinnamon, and a pinch of salt.
Cook for about 6 to 9 minutes, stirring occasionally, until the apples are tender but still hold their shape.
Continue cooking briefly if there is excess liquid. The remaining syrup should lightly coat the apples rather than pool around them.
Transfer the apples to a bowl and let them cool completely.
Add the chopped pecans to a dry skillet over medium-low heat.
Toast for about 3 to 5 minutes, stirring frequently, until fragrant and lightly browned. Remove them from the pan immediately. Add a pinch of salt if desired.
Beat the softened vegan butter for the frosting until smooth and creamy.
Gradually add the powdered sugar, beating until combined.
Add the maple syrup, vanilla, and a pinch of salt. Beat until smooth and fluffy.
If the frosting is too firm, add dairy-free milk 1 tablespoon at a time. If it becomes too soft, beat in a little more powdered sugar.
Spread the maple frosting evenly over the completely cooled cake.
Spoon the cooled caramelized apples over the frosting, leaving behind any large amount of thin liquid that may remain in the bowl.
Sprinkle with the toasted pecans.
If the apple pan contains a small amount of thick, syrupy liquid, drizzle a little over the cake.
Slice into 12 servings and serve.