Preheat the oven to 325 degrees F (165 degrees C). Generously grease a 10-inch fluted tube pan (such as Bundt).

Make the cake: Beat white sugar and butter together in a large bowl using an electric mixer until light and fluffy. Add eggs one at a time, beating well after each addition. Stir in flour, 1 cup at a time, blending well after each addition until batter is smooth; stir in lemon extract. Mix lemon-lime beverage into batter and pour into the prepared pan.

Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, about 1 hour 15 minutes. Remove cake from the pan immediately by placing a wire cooling rack over the base of the cake and inverting the pan to release.

Make the glaze: Beat confectioners' sugar, milk, and vanilla together in a bowl using an electric mixer on medium until glaze is smooth. Pour glaze over warm cake; let cool completely.