Preheat the oven to 350°F (175°C). Grease and flour a 9-inch cake pan or bundt pan. In a large bowl, cream together the butter and sugar until light and fluffy. Add eggs one at a time, mixing well after each addition. Stir in the lemon zest and vanilla extract.
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Gradually add the dry ingredients to the butter mixture, alternating with the buttermilk and lemon juice. Mix until just combined.
Pour the batter into the prepared pan and spread evenly. Bake for 35 minutes or until a toothpick inserted into the center comes out clean. Allow the cake to cool in the pan for 10 minutes before transferring it to a wire rack.
While the cake is still warm, heat the lemon juice and sugar in a small saucepan over medium heat until the sugar dissolves. Poke small holes in the cake with a toothpick and slowly pour the syrup over it. Let the cake absorb the syrup completely.
Once the cake has cooled, whisk together the powdered sugar and lemon juice to form a smooth glaze. Drizzle over the cake for extra sweetness and tang.