Mix butter, sugar, and egg. Stir in flour, cream of tartar, and walnuts until blended.
Divide dough in half. Stir red cherries into one half and green cherries into the other half. Shape each half into a roll, 1-1/2 inches in diameter. Wrap each in plastic wrap and refrigerate at least 4 hours.
Heat oven to 375 degrees F. Using a sharp knife, cut rolls into ¼-inch slices. Place slices 1-inch apart on parchment-lined or ungreased cookie sheets.
Bake until set but not brown, about 8 to 10 minutes. Remove from baking sheet to wire racks and cool.
Store cookies in airtight containers or freeze up to 6 weeks. Makes about 3 dozen cookies.
Recipe by The Culinary Cellar at https://theculinarycellar.com/sphere-magazine-december-1973/