For filling, in a medium saucepan, stir together rhubarb, strawberries, the 1/2 cup granulated sugar, and the 1/4 cup cold water. Cook, covered, about 5 minutes or until rhubarb is tender. Stir together the 2 tablespoons water and the cornstarch. Stir into rhubarb mixture. Cook and stir until thickened and bubbly. Set aside to cool.
For dough, in a large bowl, beat butter with an electric mixer on medium to high speed for 30 seconds. Add brown sugar and the 1 cup granulated sugar. Beat until fluffy. Add eggs and vanilla. Beat until combined. Add flour, baking soda and salt; beat well.
Divide dough in half. Roll one-half of the dough between sheets of waxed paper to a 12x10-inch rectangle. Remove top sheet of waxed paper. Spread half of filling on dough. Beginning at one long side, roll up and seal ends and edge of dough. Wrap in waxed paper or plastic wrap. Repeat with remaining dough and filling. Chill in the refrigerator for at least 2 hours or up to 24 hours or freeze dough.
If necessary, reshape the log of dough before slicing. Cut dough into 1/4-inch-thick slices. Place 2 inches apart on greased cookie sheets. Bake in a 350 degree F oven about 10 minutes. Makes about 54 cookies.