In a mixing bowl with a hand-held electric mixer, beat egg; blend in brown sugar. Combine flour, soda, and salt in a separate bowl; add to egg mixture a little at a time, alternating additions with the oil.
Stir in vanilla and peanuts. Drop by teaspoonfuls onto greased baking sheet. Bake peanut cookies at 375° for about 6 to 8 minutes.
Makes about 3 dozen peanut cookies.