In large bowl, cream sugar and butter until light and fluffy. Blend in vanilla and eggs. Lightly spoon flour into measuring cup; level off. Stir in flour, baking soda, and salt.Fold in gum drop pieces. Shape dough into two 1 1/2 inch diameter rolls.Wrap in waxed paper; chill at least 1 hour.
Heat oven to 350F. Grease cookie sheets. With sharp knife, slice into 1/4-inch slices; place on prepared cookie sheets. Bake at 350 for 8 to 11 minutes. Remove from cookie sheets immediately. Make 5 to 6 dozen.