Although there’s no reason you couldn’t use English walnuts.
From a box sold in Adams, Minnesota, with ephemera from Ohio.
Black Walnut Cookies
3 c. flour
1 c. brown sugar
1 c. butter
1 tsp. baking soda
1 tsp. cream of tartar
1/4 tsp. salt
Beat on low until well mixed. Stir in 1 c. black walnuts (chopped).
Divide dough in 1/2. Shape into two 2-by-12 inch rolls.
Chill (2 hours or more). Cut 1/4 inch slices on ungreased cookie sheet.
Bake 350 deg., 2 to 9 minutes until lightly browned.