Cream together shortening, sugar, egg and vanilla.
In a separate bowl, sift together flour, baking soda and salt.
Mix the dry with the wet ingredients.
Use a cookie scoop or a 2 Tablespoon measuring spoon to scoop and drop cookies onto a cookie sheet.
Bake at 375 degrees for 8-10 minutes. Remove from oven and cool.
Put a 1/4 cup of butter into a sauce pan over medium-medium high heat. The butter will melt and then come to a boil.
Allow the butter to cook until the bubbles have died down and you are left with a brown butter.
Add in powder sugar a little at a time. After the sugar has been mixed in, add about a teaspoon of water so the frosting becomes a spreading consistency. If the frosting is watery or it appears the sugar hasn't absorbed all of the butter, add more powedered sugar.
When the cookies have cooled, gently frost the cookies with the brown butter frosting.