Preheat oven to 350°F. Line the bottom of a 9-inch springform cake pan with a round piece of parchment paper. (Optional: line the bottom and sides of a standard cake pan with parchment. Create 2 long strips of parchment that overlap the sides of the pan. Lay them crosswise on the bottom of the pan. Place a round piece of parchment on top. These strips will allow you to loosen and slightly lift the cake from the pan before inverting it onto an oven safe platter.)
Arrange the peach slices in the bottom of the pan in an overlapping fashion so that there are no gaps.
In a medium bowl, whisk together the milk, maple syrup, vinegar, ground flaxseed, and vanilla. Set aside.
In a large bowl, whisk together the flour(s)*, baking powder, baking soda, and salt until well blended.
Add the wet mixture to the dry mixture and use a wooden spoon or spatula to mix well.
Pour the batter into the prepared cake pan. Bake 45 to 50 minutes, or until a toothpick inserted in the middle of the cake comes out clean.
Cool the cake in the pan for at least a couple of hours. Remove the sides of the pan (or run a spatula or knife around the sides of the pan and pull on the corners of the parchment to loosen the cake). Invert the cake onto an oven-safe platter and remove the parchment paper.
If the top of the cake is soggy, bake at 325°F for an additional 30 minutes or until dry.