Preheat oven to 350°F, and grease a 9×13-inch pan with a nonstick spray or line with parchment paper.
Add lemon cake mix and pudding mix to a medium bowl. In another bowl, combine all wet ingredients, eggs and the buttermilk. Add to dry mixture and beat until combined.Pour into a 9×13 pan.
Bake for 25-35 minutes (or for more slices and thinner cake, use an 11×15 pan and bake for 18-20 minutes. Stick a toothpick in the center to check for doneness.
It may require a couple minutes longer baking time for because of the extra moisture produced by the milk.
While the cake is cooling, make the icing, mix powdered sugar, lemon juice, oil and with a mixer until smooth. Add one more tablespoon of water if consistency isn't saucy.
Pour a THIN layer on top of the cake when just a little cooled.
