For the cake: Preheat the oven to 350 degrees F. Grease and flour an 8-inch square baking pan. Line the bottom with parchment and spray with cooking spray.
Pulse the red cabbage in a food processor until pureed. Add the sugar, apple juice and eggs and process until well combined. With the motor running, add in the oil and process until combined and emulsified.
Combine the flour, baking powder, salt and cinnamon and whisk to combine. Add to the food processor and pulse until just combined. Pour into the prepared pan.
Bake until a toothpick inserted in the center comes out clean, 30 to 35 minutes. Let cool. Turn out onto a plate and remove the parchment.
For the frosting: In a mixer fitted with a whisk, beat together the Greek yogurt and honey. Frost the cake and serve.