Preheat oven to 300 F. Line inside of two 9 inch round metal cake pans with nonstick or line with regular foil and rub with butter.
In a large mixing bowl, beat the butter and both sugars until creamy using medium high speed of an electric mixer. Beat in the vanilla. Scrape sides of bowl and beat in egg. Beat just until smooth.
Beat in the salt and baking soda. Scrape sides of bowl again.
With a mixing spoon, gradually stir in the flour. When flour is incorporated, stir in the chips.
Divide batter in half and place each half in center of cookie pan. With a soft spatula or scraper, press into circles, leaving about a 1/2 inch border between the rim of the pan and the dough (cookie will spread as it bakes).
Bake side by side on center rack for about 30-35 minutes or until cookies appear set. Let cool completely in the pan, then lift from pan and decorate with icing.
To make icing, mix the butter and sugar together in a mixing bowl. Beat on low with an electric mixer, gradually increasing speed and scraping sides of the bowl. Beat in the vanilla and add milk or cream one tablespoon at a time until you get the consistency you want. Add a pinch of salt to pick up the flavor (only if using unsalted butter and/or shortening).