1For the Apples -
Peel & core apples cut into a 1/2 - 1 inch dice.
Melt butter in a saute pan, after butter has melted add sugars and cook over med heat just after you reach carmelization (thick and sticky). Add in apples, cinnamon, nutmeg, and Lemon Juice continue to cook apples until soft and sugar mixture reaches a thick consistency. (The apples will thin out the sugar when you add them in so you will want to cook out as much moisture as possible). Drain and set aside. The apples should be completely cooled before you add them to the cookie batter.
You may also want to save the sugar mixture from the apples it makes a wonderful glaze, you may also add a teaspoon of vanilla when the apples are done for some added flavor if you like.
2For the browned butter. Place 2 sticks of butter in a sauce pan over med high heat, cook butter until it develops a light amber brown color. You will need to keep an eye on this and be sure to stir a few times. It takes approx 10 mins to develop the color. Don't be alarmed when you see foam and bubbles this is normal just stir and continue. When butter reaches its proper color remove from the heat into a heat proof container to cool. (leaving it the pan will burn the butter as it will continue to cook.) butter should smell very nutty.
3Cream together butter, cream cheese, sugars, and vanilla until fluffy.
4Add eggs one at a time.
5Sift together all dry ingredients. Add to creamed mixture.
6Fold in cooled apples. (I make large cookies) Place large spoonful's of batter onto cookie sheet. Bake at 375F for 18 minutes. If you prefer smaller cookies bake for 12 minutes.
When I bake cookies I use a double insulated pan as they never burn and always come out with beautiful color. Always remember cooking times and ovens vary greatly.
7For the brown butter icing -
In a mixer add all ingredients but the regular milk and mix on med speed until combined (NOTE: mixture will look very crumbly) continue mixing and add milk until you reach your desired consistency. Using a piping bag, cornet (if you have parchment and know how to make one) or a plastic bag cut a small hole on one of the bottom corners, pipe icing onto cooled cookies.