Melt butter in a medium sized pot and allow to cool until room temperature or slightly warm.
Stir sugars, egg, and vanilla into butter in medium pot until smooth.
In a separate bowl, combine dry ingredients (flour, oats, baking soda, and salt).
Stir dry ingredients into butter mixture just until combined (over stirring could impact the final product). The mixture will be a little thin.
Let the dough rest for about 10 minutes. The dough will thicken into a nice consistency to roll into balls. Use 2 tablespoons of dough for each dough ball. Refrigerate dough balls in a sealed container for an hour for thinner, crispier cookies (still a little chewy in the middle). Refrigerate overnight for thicker, chewier cookies.
On a baking sheet covered with parchment paper, bake at 325°F for 14-15 minutes or until just golden brown on the edges. The cookies will spread (the ones refrigerated for an hour will spread more than those kept in the fridge overnight). Don't worry - thin cookies make a lovely cookie sandwich!! (So do the thicker ones!) But that does mean to give each dough ball plenty of space on the cookie sheet.
Immediately after removing cookies from the oven, follow instructions for Achieving the Perfectly Round Cookie. This step is super important to create uniform cookies that meld together nicely as sandwiches. Allow cookies to cool on the cookie sheet for 10 minutes. Then remove cookies to another flat surface to cool completely. Pipe or spread Vanilla Ermine Frosting on the bottom of half of the cookies. Top with the other half of each cookie to make cookie sandwiches. Enjoy!