Stir together dates, 1/4 cup sugar, 1/2 cup water, and 1/8 teaspoon salt in a saucepan; bring mixture to a boil over medium-high heat. Reduce heat; simmer 3 to 5 minutes. Remove from heat; stir in walnuts, and set aside.
Beat remaining 1/2 cup sugar, softened butter, and brown sugar at medium speed with an electric mixer until light and fluffy.
Add egg and vanilla, beating until blended. Gradually add remaining 1/8 teaspoon salt, flour, and baking soda, beating until blended. Cover and chill 1 hour.
Turn dough out onto floured wax paper, and roll into an 18- x 12-inch rectangle. Spread date mixture evenly over dough, leaving a 1/2-inch border.
Roll up dough jellyroll fashion, beginning at 1 long side. Wrap in wax paper, and chill 1 hour.
Cut roll into 1/4-inch-thick slices; place on lightly greased baking sheets.
Bake, in batches, at 375° for 12 to 14 minutes or until lightly browned. Cool cookies 2 to 3 minutes on baking sheets, and remove to wire racks to cool completely.