1Preheat oven to 350 degrees F (175 degrees C). Grease 10 muffin cups or line with paper muffin liners.
2Beat sugar and oil together in a bowl with an electric mixer. Add eggs and lemon zest; beat until smooth.
3Stir flour, poppy seeds, baking powder, and salt together in bowl; gradually beat into the sugar mixture alternately with buttermilk and 2 tablespoons lemon juice until you achieve a smooth batter. Stir vanilla extract into the batter; pour into prepared muffin tins.
4Bake in the preheated oven until golden and the tops spring back when lightly pressed, 25 to 35 minutes. Cool muffins in pans for a few minutes before moving to a wire rack to cool completely.
5Stir confectioners' sugar and 2 tablespoons lemon juice together in a bowl; drizzle over tops of cooled muffins.