Step 1Heat the oven to 350 degrees. Lightly spray a muffin pan with vegetable oil.
Step 2In a small saucepan, bring one-half cup water, one-half cup of the milk and a pinch of salt to a boil. Add the oats, cover and simmer at medium-low heat until cooked, about 5 to 7 minutes, stirring occasionally. Most of the liquid will evaporate.
Step 3Heat the butter and 2 tablespoons of the white sugar over medium-high heat until bubbling. Add the apples and 1 teaspoon of the cinnamon, toss, then allow to caramelize without stirring. Toss again, each time allowing the apples to color. Once finished cooking, the apples should have lost their edges and be caramelized, about 5 minutes.
Step 4Into a medium mixing bowl, sift together the flour, oat flour, graham flour, the remaining 2 tablespoons white sugar, the brown sugar, baking powder, nutmeg, the remaining 1 teaspoon cinnamon and one-half teaspoon salt. Add any remaining grain from the sifter and stir to combine.
Step 5Whisk the remaining 1 cup milk, the applesauce, the melted butter, the egg and the vanilla with the oatmeal in a separate bowl. Pour in the dry-ingredients mixture. Add the apples and fold until the mixture just comes together.
Step 6Using an ice cream scoop (about one-half cup capacity), scoop the mixture into each of 12 muffin cups, about 1 scoop per muffin. Bake for 35 to 40 minutes. The bottom of the muffins will be dark golden brown.