Hi,
I have a recipe for a tea cakes that my Grandmom had in her book It is pretty old book of her recipes handed down but it has the same things in it and they are rolled and baked without chilling
They were Sugar Cookies to me when I made them so maybe they just had a different name back then??
So it Might be close?
She never put how much of anything came from the recipes Sorry If I remember it is a nice amount though as all her cookies in the one book was what she made at Christmas and well there were always plenty as she started at Thanksgiving and kept them in tins so there had to be enough for us to sneak LOL

Tea Cakes

5 cups flour
4 eggs
2 cups sugar
1 cup butter or margarine
1 cup thick buttermilk
1 teaspoon baking soda
1 teaspoon salt
1 teaspoon nutmeg
1 teaspoon vanilla

In a large bowl, place about 5 cups of flour. Make your well in the center. Put rest of ingredients in the well. Using your hand gradually stir in flour as you mix the ingredients in the well of the flour. Continue till you have a stiff dough add a little more flour if you need to.

On floured waxed paper roll dough thin cut with cookie cutters or glass put on cookie sheet don't let them touch.

Bake them at 350 degrees until edges get brown. Makes a good many cookies

I also checked around some of my favorite cookie sites Here is a recipe that may be close also it has the right ingredients.

Country Fair Sugar Cookies
These cookies are very soft and cake-like.

4 1/2 cups sifted all-purpose flour
4 teaspoons baking powder
1 teaspoon baking soda
1 teaspoon ground nutmeg
1/4 teaspoon salt
1 cup butter or margarine, softened
2 cups granulated sugar
3 large eggs
1 tablespoon vanilla extract
1 cup buttermilk

Additional granulated sugar for sprinkling (turbinado sugar is especially nice.)

Preheat oven to 350F (175C).

Mix together the flour, baking powder, baking soda, nutmeg and salt; set aside.

In large bowl cream together the butter and sugar until fluffy; add the eggs and vanilla and blend well. Add the buttermilk alternately with the flour mixture until just mixed. Let dough rest for 10 minutes.

Drop by tablespoonfuls onto greased baking sheets. Sprinkle with additional sugar.

Bake for 10 to 12 minutes. Transfer to wire racks to cool completely. Store in sealed container.

Makes 4 dozen.