Cream butter in a large mixing bowl; gradually add sugar, beating well. Add eggs and vanilla; beat well. Combine flour and soda in a small mixing bowl; add to creamed mixture, mixing well. Stir in oats and peanuts.
Shape into 2 1/2-inch balls; place 3 inches apart on greased cookie sheets. Flatten cookies with a fork in a crisscross pattern. Bake at 375° for 10 to 12 minutes. Cool slightly on cookie sheets; remove cookies to wire racks, and cool completely.