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 butter and granulated sugar together with an electric mixer 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 to a wire rack to cool completely.
While the cake cools, prepare the lemon yogurt topping by whisking together the powdered sugar, plain Greek yogurt, fresh lemon juice, and lemon zest in a small bowl until smooth.
Once the cake is completely cool, spread the lemon yogurt topping evenly over the top of the cake.
Slice and serve.