I had never heard of a tea cake until I was married. My wife raved and raved about her Grandmother's tea cakes and how delicious they were. When she was in High School she was very active in Track and Field and basketball. It drives her crazy when i brag about how well she did in athletics and talk about all the awards she won. Her Grandmother would make a huge batch of tea cakes for her to take on the out of town meets. Her coach always asked if Margaret's tea cakes were, "coming with us."
Tea Cakes are a different kind of cookie in that they aren't really sweet. They are a dense and soft cookie slightly sweet and very subtle in flavor. I think they would be really good with some lemon zest and a lemon glaze. But, these weren't for me, they were for my wife who likes them just the way they were. I did tweak the recipe adding a little more baking soda, vanilla, and almond extract. I liked they way they turned out.
A word of caution though, if you don't like to roll of fiddle with cookies, walk away. The first dozen I made I just dropped out of my scoop. They didn't look very pretty at all, very rough. The amount of flour in the batter made them hold their shape and not spread as I thought they would. I rolled them in my hands and patted them out (that's what made them look so smooth). You could roll the dough out and cut them, but I didn't think that step was necessary.
1 cup butter
2 cups sugar
3 eggs
6 tablespoons of milk
5 1/2 cups flour
1 Tablespoon baking powder
1/2 teaspoon salt
1/8 teaspoon baking soda
1 tablespoon vanilla
1/8 teaspoon almond extract
Cream butter and sugar together until light and fluffy (about 5 min.). Add eggs, milk, and extracts mixing to incorporate. Slowly add flour until completely combined. Drop by rounded spoonfuls onto a parchment lined cookie sheet, rolling into a ball and slightly flattening out. Bake in a 350 degree preheated oven 10-12 minutes. They will not brown. Cool on a wire rack. Store in an airtight container.
Note: I think they would be delicious with some lemon zest added to the dough, then glazed with a lemon glaze.