Mix the dry ingredients: Combine 1⅓ cups unbleached, all-purpose flour, ½ teaspoon baking powder, and ½ teaspoon salt in a separate bowl. Set aside.
Cream the Butter and Sugar: In a large mixing bowl, whip ¼ cup butter until smooth. Add 1 cup sugar and continue beating until the mixture is light and fluffy. Add 1 large egg and mix well.
Add Lemon and Yogurt: Stir in grated rind of 1 small lemon, 1½ tablespoons lemon juice, and ¾ cup Greek yogurt.
Combine and Chill: Add dry ingredients to the wet mixture and mix until smooth with no dry spots. Cover and refrigerate for at least 30 minutes to prevent excess spreading and help create the soft, cake-like texture.
Bake the Cookies: Drop spoonfuls (about 1 tablespoon each) onto a parchment-lined baking sheet. Bake at 375˚F (190˚C) for 12–15 minutes, or until the bottoms are golden and the edges begin to brown.
Cool: Let cookies cool on the pan for 10 minutes. Then transfer to a rack or towel to finish cooling before glazing.
Make the Glaze: Sift 1 cup powdered sugar into a small bowl. Add grated rind of 1 lemon, 1 tablespoon of heavy cream, and 1-2 tablespoons lemon juice. Combine until smooth. Add another tablespoon of lemon juice if the icing is too thick.
Glaze the Cookies: Frost warm cookies lightly. Let icing set before serving or storing.