Preheat the oven to 400ºF. Spray a 12 cup muffin pan with cooking spray or line with paper muffin cups.
In a large bowl, rub together the sugar and lemon zest with your fingertips until the sugar is moist and fragrant. Whisk in the flour, baking powder, and baking soda. In another bowl whisk together the sour cream, eggs, vanilla, lemon juice and butter. Add the wet ingredients to the dry and, with a rubber spatula, gently stir just until combined. Stir in the poppy seeds. Divide the batter evenly amongst the muffin cups.
Bake for 18-20 minutes or until the tops are golden and a toothpick inserted in the center of a muffin comes out clean. Transfer the pan to a rack to cool for 5 minutes. Remove the muffins and let them cool completely on the rack before icing them.
To make the icing, place the confectioners' sugar in a small bowl and add 1 tablespoon of milk. Stir with a spoon until the milk is absorbed, then add additional milk ½ teaspoon at a time until the icing is thin enough to drizzle off the spoon. Drizzle lines of icing over cool muffins.