Preheat the oven to 350°F (175°C). Grease and flour an 8-inch round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, beat the softened unsalted butter and granulated sugar with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the plain yogurt, fresh lemon juice, lemon zest, and vanilla extract until combined.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not overmix.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes. Then, transfer the cake to a wire rack to cool completely.
While the cake cools, prepare the lemon yogurt topping by combining the powdered sugar, plain yogurt, fresh lemon juice, and lemon zest in a small bowl. Stir until smooth and well combined.
Once the cake is completely cool, spread the lemon yogurt topping evenly over the top of the cake.
Slice and serve the lemon cake with lemon yogurt immediately or refrigerate until ready to serve.