Position a rack in the center of the oven. Preheat the oven to 325º. Coat a 9-by-5-inch metal loaf pan with nonstick cooking spray. Line the pan with an 8-1/2-inch-by 15-inch strip of parchment paper so it covers the bottom of the pan and hangs over the long sides.

Put the sugar in a small bowl. Using a microplane zester, grate the zest from the lemons into the sugar. Rub the zest into the sugar with your fingertips until the sugar is sandy and moist and fragrant with lemon. Set aside.

Cut the lemons in half. Using a citrus reamer, squeeze the juice from the lemons into another small bowl.

In a third small bowl, whisk together ¼ cup of the fresh lemon juice, (you should have about 2 tablespoons remaining; set aside for the icing.) the sour cream, vanilla, and lemon oil. Set aside.

In a medium bowl, sift together the flour, baking powder, and salt.

In a stand mixer fitted with the paddle attachment, beat the butter and oil together on medium speed just until creamy. Add the lemon sugar and beat until light and fluffy. Add the eggs, one at a time, beating well after each addition and scraping down the bowl as you go. Decrease the mixer speed to low and beat in the sour cream mixture. Add the dry ingredients in two additions, beating on low for a few seconds after each.

Scrape the batter into the prepared pan and smooth the top with a spatula. Bake for 55 to 65 minutes (do not check the cake until it has been in the oven for at least 40 minutes, as it might sink in the middle.) The cake is done when a wooden skewer inserted into the center of the cake comes out clean. Transfer the loaf, still in the pan, to a wire rack to cool 10 minutes. Grasping the overhanging parchment, lift the loaf out of the pan and set it on the rack. Cool completely.

To make the icing, whisk the butter, 2 tablespoons lemon juice, and lemon oil with the confectioners’ sugar until smooth and creamy. Spread or drizzle the icing over the cooled cake.