Let your butter soften at room temperature for a couple hours. Wash, zest, and juice 2-3 lemons. Spray a 12 cup Bundt pan with Baker's Joy cooking spray, or use shortening and flour. Preheat your oven to 350°.
Combine butter, oil, and sugar in a large mixing bowl. Beat wtih an electric mixer at medium speed for about 5 minutes, or until light and fluffy.
Add the eggs, vanilla, lemon zest, and juice. Beat until eggs are thoroughly combined, then beat in the milk.
Whisk or sift in the flour, dry pudding mix, soda, and salt. Stir until well mixed.
Spread the batter in the prepared pan. Bake at 350° for about 60-65 minutes, or until a toothpick inserted in the center comes out with moist crumbs, but no batter.
About 5 minutes before the cake is done, prepare the icing. Melt the butter in a small glass bowl. Add the powdered sugar, salt, zest, and enough lemon juice to make a thin glaze.
Remove the cake from the oven and brush the crust with some of the icing. Let cool in the pan for about 15 minutes, then carefully invert onto a cooling rack.
Brush the remaining icing on the top and sides of the cake. Let cool for at least 20 more minutes, or until the glaze sets.
Serve cake warm or at room temperature.