Your own homemade chunky applesauce is the key to this flavorful cake. That way, the cake is just as much (if not more) about the fruit than the batter.
Place the sliced apples in a saucepan with just enough water to keep the bottom of pan moist. Cook over medium heat, stirring frequently, until the apples are about half broken down into sauce and the rest tender, leaving the mixture chunky. How long this takes depends very much on the variety of apple, so check frequently. Remove from the heat. This step can be done ahead of time.
Preheat the oven to 350 degrees F.
Combine the flour, baking powder, baking soda, and 1 teaspoon of the cinnamon in a large mixing bowl and mix thoroughly.
Make a well in the center of the dry ingredients and pour in the syrup, vanilla, and oil. Stir together until the flour mixture is moistened. It will barely hold together as a batter.
Stir in the chunky applesauce, walnuts, and the remaining cinnamon and stir briefly, just until evenly distributed in the batter.
Add just enough non-dairy milk so that the flour mixture is completely moistened, but let the batter remain very stiff.
Scrape the mixture into a lightly oiled deep 9-inch round cake pan or a 9 by 12-inch baking pan. Pat the mixture down lightly with the back of a spatula, as it will be rather lumpy.
Bake for 30 to 35 minutes, or until golden and a knife inserted into the center tests clean, other than any moisture from the apples. Allow to cool, then cut into wedges or squares to serve.