Preheat oven to 325° Grease a 9x5" loaf pan and line the bottom with parchment paper. Set aside.
Combine sugar, butter, lemon juice, lemon zest, and vanilla in a Kitchenaid or large mixing bowl. Beat for about 3 minutes, or till very creamy.
Beat in the sour cream. Beat in the eggs one at a time till mixture is well blended.
In a small bowl, whisk together the flour, cornstarch, baking powder, and salt. Add to the wet ingredients and blend well.
Pour batter into the prepared loaf pan. Bake at 325° for 60-70 minutes or till a toothpick inserted in the middle comes out clean. Cool for 20 minutes.
While cake is cooling, combine sugar and lemon juice in a small saucepan. Bring to a boil and simmer for about 5 minutes. Poke holes in the cake with a toothpick, then brush the glaze over the tops and sides of the cake.
Cool cake completely, then drizzle with lemon icing.
For icing, whisk together all ingredients till smooth, adding enough milk as necessary to make a thin consistency.