On medium speed, in large bowl, blend sugars; add butter, mix till grainy consistency. Add eggs, peanut butter and vanilla extract; beat till fluffy. On low speed, add flour mixture until barely mixed into butter mixture.
Drop by rounded tablespoonsful, 2 inches apart on ungreased cookie sheet. Use a wet or flour coated fork to gently press a crisscross pattern on top of each cookie.
Bake until the edges of the cookies are lightly browned, approximately 18 to 20 minutes. Remove to cool on a flat surface.