Put the sliced apples in a medium saucepan with the water, which should be just enough to keep the bottom of the pan moist. Cook over medium heat, stirring often, until about half of the apples have broken down into sauce and the rest are tender but still in pieces, 10 to 20 minutes depending on the variety. Add a splash more water if the pan gets dry before the apples soften. Remove from the heat. (The applesauce can be made up to 3 days ahead and refrigerated.)
Make the cake
Preheat the oven to 350°F (175°C). Lightly oil a deep 9-inch round cake pan or a 9 x 13-inch baking pan.
In a large mixing bowl, combine the flour, baking powder, baking soda, and 1 teaspoon of the cinnamon, and mix thoroughly.
Make a well in the center of the dry ingredients and pour in the maple syrup, vanilla, and oil. Stir until the flour mixture is moistened. It will barely hold together at this point.
Add the chunky applesauce, walnuts, and the remaining 1 teaspoon cinnamon, and stir briefly, just until evenly distributed.
Stir in the nondairy milk, 1 tablespoon at a time, just until no dry flour remains. The batter should stay very stiff and lumpy.
Scrape the batter into the prepared pan and pat it down lightly with the back of a spatula.
Bake for 30 to 35 minutes (start checking a 9 x 13-inch pan at 25 minutes), until the top is golden and a knife inserted into the center comes out clean, apart from any moisture from the apples. Let the cake cool in the pan for at least 30 minutes, then cut into wedges or squares.
Notes
Make ahead: The chunky applesauce can be cooked up to 3 days ahead and kept covered in the refrigerator.Apples: Soft varieties such as McIntosh break down in about 10 minutes; firm ones such as Honeycrisp or Granny Smith can take 20 minutes or more. A mix of the two gives both sauce and chunks.Gluten-free: Replace the whole wheat pastry flour with a 1-to-1 gluten-free baking flour, such as Bob's Red Mill.Pan size: A 9 x 13-inch pan makes a thinner cake; start checking it at 25 minutes.Storage: Keep covered at room temperature for 1 day, refrigerate up to 4 days, or freeze wrapped slices up to 3 months.