Preheat oven to 325F. Grease bottom of a ⅛ sheet pan.
Sift the flour with the baking soda and whisk in the salt. Set aside.
In a mixing bowl, preferably a stand mixer with the paddle attached, beat the butter, sugar and lemon zest until creamy. With the mixer going, drizzle in the oil.
After you've added the oil, stop the mixer and stir in one egg until it is incorporated. Beat with the mixer for 30 seconds, scraping the sides of the bowl, then repeat with the second egg. Beat until smooth, then stir in half of the flour mixture and the sour cream. Beat on low until smooth, then add remaining flour mixture, lemon juice and lemon oil (or extract).
Pour into the pan. Drop pan lightly on counter to bring air bubbles to top. Bake at 325 for about 35 to 38minutes. Definitely check around the 30 minute mark, especially if you've changed the pan size. Mine took 38 minutes.
Remove from the oven and let cool.
Lemon Frosting
Beat the softened butter until creamy. Beat in 1 cup of the powdered sugar, stopping to scrape the side of the bowl. Beat in the softened cheese, zest and half of the lemon juice. Add lemon oil or extract. and Stir.
Add remaining sugar and lemon juice as needed to desired sweetness and consistency, beating until smooth. Add a drop of yellow gel food coloring if you'd like a deeper yellow or leave it as is.