Preheat oven to 350 degrees F. Line a 12-cup standard muffin tin with paper liners; set aside.
In bowl of an electric mixer fitted with paddle attachment, beat cream cheese, sugar and pumpkin pie spice on medium speed until light and fluffy, approximately 2-3 minutes; set aside.
In a large bowl, combine flour, pumpkin pie spice, baking powder, cinnamon, baking soda, nutmeg and salt.
In a large glass measuring cup or another bowl, whisk together pumpkin puree, sugars, vegetable oil and eggs.
Pour mixture over dry ingredients and stir using a rubber spatula just until moist.
Scoop a spoonful of batter evenly into the muffin tray. Drop 2 teaspoons cream cheese filling into the center of each cup. Top with remaining batter to completely cover the filling. Place into oven and bake for 15-18 minutes, or until a tester inserted in center comes out clean. Remove from oven and cool completely on a wire rack before frosting.
In bowl of an electric mixer fitted with paddle attachment, beat butter and confectioners’ sugar on medium speed until light and fluffy, approximately 1-2 minutes. With mixer on low speed, add milk, vanilla and pumpkin pie spice, beating just until incorporated. Increase speed to medium high and beat until frosting is light and fluffy, approximately 1-2 minutes.
Use a small offset spatula or a pastry bag fitted with decorative tip #807 to frost the cupcakes, garnished with graham cracker crumbs, if desired.