Preheat your oven to 350°F (175°C). Grease and flour a loaf pan.
In a small bowl, combine the sugar with the zest of one lemon. Use your fingertips to rub the zest into the sugar until fragrant and well combined, about 3-5 minutes.
In a large mixing bowl, beat the eggs and sugar together until pale and creamy. Add the vanilla essence and mix until well combined.
Gradually pour in the oil and warm milk while continuing to mix until smooth.
Sift the flour and baking powder into the wet mixture. Gently fold the dry ingredients into the wet mixture until just combined.
Pour the batter into the prepared loaf pan, spreading it out evenly.
Place the pan in the preheated oven and bake for approximately 40-45 minutes, or until a toothpick inserted into the center comes out clean.
Once baked, remove the cake from the oven and let it cool in the pan for a few minutes. Then, transfer the cake to a wire rack to cool completely.
If desired, you can prepare a simple lemon glaze by mixing powdered sugar with fresh lemon juice until smooth. Drizzle the glaze over the cooled cake before slicing and serving.