Preheat the oven to 350℉. Grease a loaf pan or cake pan and line the bottom with parchment paper.
In a large bowl, combine the ½ cup plain full-fat yogurt, 1 cup granulated sugar, and 3 large eggs, stirring and whisking until well blended. Add the 1 ½ cups all-purpose flour, 2 teaspoons baking powder, and ¼ teaspoon fine sea salt, mixing just to combine. Add the ½ cup coconut oil and stir well to incorporate. Keep stirring until it forms a smooth batter. Pour and scrape the batter into the prepared pan.
Place in the oven and bake for 30-40 minutes until a toothpick inserted into the center comes out clean. Do not overbake.
Cool cake on a rack for about 20 minutes; then turn it out of the pan to cool completely.
When the cake is thoroughly cooled, juice the 1 lemon into a bowl and whisk in the sifted ½ cup powdered sugar. Then, evenly spoon the glaze over the top of the cake; it will soak right in.
Notes
Choose your pan. I used a standard-size loaf pan to bake my cake, and I also love using a bundt pan for a more decorative option. It also bakes up beautifully in a round 8 or 9-inch cake pan.
What type of yogurt? I prefer a whole milk plain yogurt (not Greek yogurt) in this cake. For a touch of extra flavor, you could use vanilla yogurt. You could also use low-fat yogurt, but I would not use non-fat because you'll lose out on the richness from the higher-fat yogurt.
Add vanilla extract. In keeping with the simple spirit of this recipe, I don't add vanilla extract to the batter. But, if you wish, add 1 teaspoon of vanilla extract.
Another oil. I use melted coconut oil in this cake, and I also use vegetable oil. You can use any neutral oil of choice (I don't recommend olive oil because it might overpower the delicate lemon flavor of the cake).
Make it extra lemony. If you love a lot of lemon flavor, rub the zest of 1 lemon into the 1 cup of sugar first to release its oils, then add the rest of the ingredients.
Try another citrus. Use this base recipe as your jumping-off point for other citrus flavors, like orange or lime.