Beat butter at medium speed with an electric mixer until creamy; add sugars, beating well. Add eggs and vanilla, beating until blended.
Process oats in a blender or food processor until finely ground. Combine oats, flour, and next 3 ingredients; add to butter mixture, beating well. Stir in chopped candy, white chocolate, and peanuts. Chill dough 1 hour, if desired.
Shape dough into 1 1/2" balls, and place on lightly greased or parchment paper-lined baking sheets. Chill briefly, if desired. Bake at 375º for 10 to 11 minutes or until lightly browned. Cool 1 minute on baking sheets. Remove to wire racks to cool.