In a medium sized bowl, mix the flour, salt, and baking powder.
In a large bowl or mixer, cream the butter and oil together. Add the sugar a little at a time. Mix until light and fluffy.
Add the eggs one at a time, beating after each addition
Starting with the dry ingredients, alternate adding dry ingredients and milk while mixing slowly.
Add half of the lemon juice, lemon zest (save some for garnish), and vanilla.
Pour batter into greased pan and bake for 1-1½ hours or until a toothpick comes out clean. While cake is baking, make the glaze.
To make the glaze
In a small bowl, mix together lemon juice and powdered sugar until it makes a thin glaze. If it is too thick add water a teaspoon at a time.
Once the cake is done, let cool on a wire rack for 10 min then use a toothpick to poke holes in the top of the cake.
Pour the glaze evenly over the cake then let it cool completely.
To make the icing
Mix the powdered sugar, milk, and vanilla together in a bowl. If the icing is too thick add milk, if it is too thin add powdered sugar. It should be the consistency of honey.
Pour the icing over the cooled cake and garnish with remaining lemon zest.