Melt the butter and set aside to cool. Place oats in a medium-sized bowl. Stir in both sugars and the salt and blend to break up any lumps.
In a separate bowl, whisk together the melted butter, vanilla and egg. Stir this into the oat mixture, mixing with your hands or a large wooden spoon to work in the ingredients. Form into a large ball of dough.
Chill the dough 20 minutes.
Preheat oven to 350ºF. Meanwhile, prepare a baking sheet by lining with parchment paper.
Form the chilled dough into 1-inch balls and flatten down very slightly on the baking sheet. Bake for 13 to 17 minutes or until the edges are nicely browned and the tops of the cookies are medium golden in color.
Cool well on the baking sheet (about 10 minutes) before attempting to remove, using a metal spatula or icing knife.
The dough can also be brought to room temperature and flattened more when placed on the baking sheet to make a large, crisp and lace-like cookie. Keep the cookies refrigerated.