Preheat the oven to 350°F.
Core and peel the apple, cut into small pieces and process in a food processor until the consistency of apple sauce, scrape down the side of the food processor occasionally as necessary. This yields about 1 cup of apple sauce, I had a really large apple.
Add apple sauce, peanut butter and water to a mixing bowl, mix on medium speed until thoroughly combined.
With the mixer on low speed, slowly add the flour, mix until combine. The batter will be pretty thick but still somewhat pourable.
Oil or spray a mini cupcake pan with nonstick cooking spray, wipe off excess.
Use a small ice cream scoop or a tablespoon to divide the batter among the cups. Because this batter will not rise like regular cupcakes, you can choose to fill it all the way to the top or half way, depending on the size of the pupcakes you want to make. I filled mine about 2/3 full.
Tap the pan or smooth out the top of the batter with a spoon/spatula before putting the pan in the oven so you don’t get irregular top.
Bake for approximately 25 - 30 minutes, or longer until a toothpick inserted in the middle comes out clean.
Cool pupcakes on a wire rack.
For the frosting, cream peanut butter in a mixing bowl on medium speed, slowly add tapioca starch, one tablespoon at a time until a frosting consistency is reached.
Pipe a small amount of frosting on top of each pupcake.
Store in an airtight container and refrigerate.