Beat the butter, brown sugar, and white sugar with an electric mixer in a large bowl until smooth. Beat the first egg into the butter mixture until completely blended; beat the vanilla into the mixture with the last egg.
Whisk the flour, baking soda, cinnamon, and salt together in a separate bowl. Beat the flour mixture into the butter mixture about 1/2 cup at a time. Fold the rolled oats and butterscotch chips into the resulting dough. If the dough isn’t firm enough, mix in more rolled oats, about 1/4 cup at a time, until dough is of desired consistency.
Refrigerate the dough for 1 to 2 hours before baking.
Preheat oven to 350 degrees F (175 degrees C).
Scoop up dough by heaping tablespoonful and roll into balls. Place cookies about 2 inches apart on baking sheets.
Bake in the preheated oven until very lightly browned and cookies are set in the middle, 10 to 12 minutes. Allow cookies to cool for about 5 minutes before removing from baking sheets to finish cooling on wire rack.