Preheat an oven to 375 degrees F (190 degrees C). Butter 6 muffin cups or line with paper muffin liners.
Combine the apple pie spice and white sugar in a small bowl; roll the biscuit pieces in the mixture to coat. Divide the coated pieces between the prepared muffin cups.
Combine the brown sugar, butter, and water in a small saucepan over medium heat; bring to a boil, stirring continuously. Allow the mixture to boil until the sugar is completely dissolved, 2 to 3 minutes. Spoon the mixture over the biscuit pieces.
Bake in the preheated oven until golden, and the tops spring back when lightly pressed, 8 to 12 minutes.