In a stand mixer, beat together butter, walnut oil, sugar, and almonds on medium high speed until the mixture becomes light and fluffy. Continue to beat and add eggs, a little at a time, until well incorporated. With a rubber spatula, fold in the walnuts, flour, vanilla, lemon zest, and salt.
Pour batter into a parchment paper lined 9 inch cake pan. Level the batter with a spatula, then arrange plum pieces in concentric circles over the top.
Bake cake for 70-80 minutes, until center is done. Remove from oven and let cool slightly.
Meanwhile, make the icing. Whisk together powdered sugar and lemon juice until a pourable icing forms. Adjust lemon juice and sugar as needed. When cake is slightly cooled, drizzle icing all over the top. Let cake completely cool before serving.
Recipe by Things I Made Today at https://www.thingsimadetoday.com/2015/03/20/plum-walnut-cake/