Preheat oven to 375 degrees F. In a large mixing bowl beat butter with an electric mixer on medium to high speed for 30 seconds. Add sugar, brown sugar, baking soda, and salt. Beat until combined, scraping sides of bowl occasionally. Beat in eggs and vanilla until combined. Beat in as much of the flour as you can with the mixer; stir in any remaining flour and the butterscotch pieces.
Drop dough by rounded teaspoonfuls about 2 inches apart onto ungreased cookie sheets. Bake for 9 to 10 minutes. Rotate cookie sheets halfway through bake cycle. Remove from cookie sheets immediately and cool on wire racks. Makes 4 dozen cookies.