Preheat oven to 350°.
Cream the butter, brown sugar and granulated sugar. Add egg white and vanilla and stir. In a separate bowl, mix oats, salt, baking soda and flour. Add dry ingredients to the butter mixture and mix just until blended. Fold in nuts, if using.
Spoon ½-tablespoon dough (about the size of a cherry) onto a parchment paper-lined cookie sheet. Leave a few inches between each ball; these cookies spread. Bake until edges are brown and center is still light in color, about 8 minutes.
Let cool some before removing cookies to a cooling rack. Make Brown Butter Cream Cheese Filling.
When cool, pair like-size cookies. Spread flat side of one cookie with filling (about 1 teaspoon). Top with second cookie.
If not serving right away, store cookies in the refrigerator. Leave cookies out about 20 minutes before serving to take the chill off.
Brown Butter Cream Cheese Filling
Cream all ingredients together.