Sherri, Are you looking for alternative ingredients because you don't eat white flour or milk products? or is it because you don't have the right ingredients on hand?
The cookie recipes I found contained cake flour to make them soft.
I've included one that has cake flour but not dry milk.
Hope it helps.

Black and White Cookies

New York black and white deli cookies.
Makes 2 dozen

Ingredients:

1 3/4 cups white sugar
1 cup butter or margarine, unsalted
4 eggs
1 cup milk
1/2 teaspoon vanilla extract
1/4 teaspoon lemon extract
2 1/2 cups cake flour
2 1/2 cups all-purpose flour
1 teaspoon baking powder
1/2 teaspoon salt
4 cups confectioners' sugar
1/3 cup boiling water (may need up to 1/2 cup)
1 ounce bittersweet chocolate

Directions:
1 Preheat oven to 375 degrees F (190 degrees C). Butter 2 baking sheets.
2 Combine sugar and butter. Mix until fluffy. Add eggs, milk and extracts. Mix
until smooth.

3 Combine cake flour, all-purpose flour, baking powder and salt. Add to butter
mixture, about 1 cup at a time. Combine well.

4 Using soup spoon, drop spoonfuls of the dough 2 inches apart onto baking
sheets. Bake until edges begin to brown, about 20 to 30 minutes. Cool
completely.

5 To make Frosting: Place confectioners sugar in large bowl. Add enough boiling
water until mixture is thick and spreadable.

6 Remove half of the frosting to the top of a double boiler set over simmering
water. Add the chocolate. Warm mixture until the chocolate is melted and
smooth. Remove from heat.

7 With brush, coat half the cookie with chocolate frosting and the other half with
the white.