Preheat oven to 350 degrees. Grease a 5x9 inch loaf pan and line with a parchment sling.
In a large bowl, sift together the flour, baking powder, baking soda and salt. Mix in lemon zest and set aside.
In a medium bowl, mix together the lemon juice, melted butter, vegetable oil, and buttermilk. Add in sugar and mix well. Add in eggs and vanilla, mixing until well blended.
Pour wet ingredients over dry and mix until flour is just absorbed.
Pour batter into loaf pan and bake on middle rack for 50-55 minutes or until a toothpick comes out mostly clean with just a few moist crumbs.
Allow loaf to cool in pan for 10 minutes.
While loaf is cooling, mix together juice of one lemon and 3 tablespoons of powdered sugar.
After 10 minutes, turn loaf out onto cooling rack. Using a toothpick, poke several holes in the top of the loaf. Poor glaze over the loaf and set aside to cool completely.
Once loaf is completely cool, combine 1 cup of powdered sugar, 1 tablespoon of lemon juice and 1 tablespoon of milk. The mixture should be thick, but pourable. Pour over cooled loaf and allow icing to set before slicing and serving.
Notes
** To substitute clabbered milk for buttermilk, add half a tablespoon of vinegar to a liquid measuring cup. Fill to half cup line with regular milk and set aside for 10 minutes.