Combine flour, cream of tartar, and baking soda in a medium bowl. Set aside.
Beat butter or margarine, granulated sugar, brown sugar, and oil in a large mixing bowl with an electric mixer until combined. Add egg and beat on medium speed for 3 minutes more. Add flour mixture; beat on low speed until combined. Stir in sunflower kernels. Drop by rounded teaspoons 2 inches apart on an ungreased cookie sheet.
Bake in a 375 degree F oven for 10 to 12 minutes or until edges are lightly browned. Transfer to wire racks to cool. Makes about 40.