In a large bowl, with a mixer, beat 6 tablespoons butter and the granulated sugar until well blended. Add ricotta, egg, and 1/2 teaspoon lemon extract; beat until well mixed.
In a medium bowl, mix flour, baking powder, baking soda, and salt. Add to butter mixture, stir to mix, then beat until well blended.
Butter 14- by 17-inch baking sheets. Drop batter in 1-tablespoon portions about 1 1/2 inches apart on sheets.
Bake cookies in a 325° oven until golden brown, 20 to 25 minutes. If baking more than 1 pan at once, switch pan positions every 10 minutes. With a wide spatula, transfer cookies to racks.
Melt 2 tablespoons butter. In a deep bowl, mix butter, powdered sugar, 1/4 teaspoon lemon extract, and milk until smooth. One at a time, hold cookies by the edges and dip rounded tops into icing; set in a single layer, icing up, on racks and let stand until icing is firm, about 10 minutes. Serve, or store airtight up to 1 day; freeze to store longer.