Preheat oven to 350°F. Line a 12-cup muffin tin with paper liners or spray with nonstick baking spray.
In a bowl, whisk together flour, oats, baking powder and baking soda.
In a second bowl, whisk together applesauce, vegetable oil, honey and eggs.
Add the wet ingredients to the dry ingredients and mix just until evenly moistened. Gently fold in the apple and cheddar cheese.
Evenly divide the batter between the cups of the prepared muffin tin. They should be almost full. Bake in preheated oven for 18-20 minutes, until a tester inserted in the center comes out clean.