Position racks to divide the oven in thirds and preheat the oven to 350 degrees F. Lightly coat cookie sheet(s) with cooking spray and set aside.
In a small saucepan or syrup pan, melt the butter and cook over low heat until the butter begins to turn a golden brown and has a nutty aroma, about 4 minutes. Watch carefully so the butter does not burn. Set aside to cool slightly, then place in a metal bowl (to avoid thermal shock) and freeze or refrigerate about 5 minutes to solidify butter.
Whisk in a mixing bowl the cake flour, baking soda, and salt. Set it aside.
In a large bowl with an electric mixer, cream together both sugars, the solid butter, and the cream cheese. Beat until smooth. Add the egg plus white and vanilla and beat well. With the mixer on slow speed, add the dry ingredients, then the chocolate chips. The dough will be soft but not runny; it does not need to be chilled before use but can be made ahead and chilled for convenience.
Drop the dough by tablespoonfuls on the prepared cookie sheets; allow a couple of inches between cookies for spreading during baking. Bake for 11 to 12 minutes, until golden with a slightly darker edge. The longer the cookies bake, the crisper they will be. Transfer with a spatula to a wire rack; when cool, store in an airtight container.
Recipe courtesy of Let Them Eat Cake, by Susan Purdy