Prep: Preheat oven to 350°F with the rack in the middle. Grease an 8x4" loaf tin, or line with parchment for easier removal. (You can use a 9x5" loaf tin, but the loaf will be less tall and domed. You may also need to reduce cook time by 10-15 minutes.)
Make lemon sugar: To a mixing bowl, add granulated sugar and lemon zest. Use your fingertips to rub them together to release the oils from the zest. Do this for about a minute — till the sugar is super fragrant and has the texture of slightly moist sand.
Wet ingredients: Add in the plant milk, lemon juice, optional lemon extract, vanilla extract, and vegetable oil. Whisk vigorously to mix everything together, so the oil emulsifies into the wet mixture (rather than simply floating on top).
Dry ingredients: In a separate bowl, whisk together the all-purpose flour, baking powder, and salt. Add into the wet mixture and use the whisk to combine everything into a batter. It's ok if there are some small lumps, as long as there are no dry streaks of flour. Avoid over-mixing.
Bake the loaf: Pour the batter into the prepared loaf tin. Bake in preheated oven for 55-60 minutes. (If using a larger loaf tin, start checking around 45 minutes.) If the top is browning too fast, loosely tent with a sheet of aluminum foil. When done, a toothpick inserted into the center of the loaf should come out clean or with just a few moist crumbs.
Let it cool: Remove the loaf from the oven. Let it cool in the pan for about 10 minutes. Then, carefully remove it from the pan and transfer to a wire cooling rack to cool completely before icing. (The icing will not set if the loaf is warm.)
Ice the loaf: Once the loaf is cool, whisk together ingredients for the icing. Position the loaf on the cooling rack over a baking tray to catch the extra glaze. Pour icing evenly over the cooled loaf. Allow it to set for at least 20 minutes, or longer as needed to harden. Then slice and enjoy!