Preheat the oven to 350°F (175°C). Grease and flour the cake pan or line it with parchment paper.
In a large mixing bowl, combine the softened butter and sugar. Use an electric mixer or whisk to beat until the mixture is light and fluffy, about 3-4 minutes. The mixture should turn pale and slightly increase in volume.
Add eggs one at a time, beating well after each addition. The batter should become smooth and slightly increase in volume. Scrape down the sides of the bowl as needed.
In a separate bowl, whisk together the flour, baking powder, and salt. Gradually add the dry ingredients to the wet mixture, alternating with milk, beginning and ending with the dry ingredients. Fold in the lemon zest and vanilla extract with a rubber spatula until just combined. The batter will be thick but pourable.
Pour the batter into the prepared pan and smooth the top with a spatula. Place in the center of the oven and bake until the top is golden and a toothpick inserted in the center comes out clean, about 30 minutes. The cake should spring back slightly when pressed.
Remove the cake from the oven and let it cool in the pan for 10 minutes. Then, transfer to a cooling rack and cool completely before serving. The finished cake will have a light, textured crumb with a golden exterior and a hint of lemon aroma.
