Preheat the oven to 350°F.
Grease and flour the inside of a 9-inch round cake pan.
Whisk the flour, cornstarch, baking powder, baking soda, salt, and cinnamon together in a large mixing bowl.
In a small bowl or liquid measuring cup, stir together the maple syrup, oil, brown sugar, non-dairy milk, maple extract, vanilla extract, and apple cider vinegar.
Pour the maple mixture into the bowl with the flour mixture, then stir the ingredients together just until combined.
Fold in the walnuts.
Transfer the batter to the prepared baking pan, smoothing out the top with a spatula.
Bake the cake for 36 to 40 minutes, until the top is golden brown and a toothpick inserted into the center comes out clean.
Place the cake pan on a cooling rack when it comes out of the oven. Let the cake cool completely.
To make the frosting, beat the butter on high speed with an electric mixer until creamy, about 1 minute. Beat in the maple syrup, vanilla, and salt.
Turn the mixer speed down to medium, then begin beating in the powdered sugar, adding a bit at a time, until the frosting is creamy and fluffy.
Once the cake is cool, invert it onto a plate. Spread the frosting over the cake, then optionally garnish with walnut halves.
Slice and serve.