Preheat your oven to 175°C (350°F). Grease an 8-inch (20 cm) loaf pan and line it with parchment paper for easy removal.
In a large mixing bowl, rub the sugar and lemon zest together with your fingertips until the mixture is fragrant and aromatic. Add the milk, lemon juice, yogurt, vanilla, and oil to the bowl, then whisk until the liquid base is completely smooth.
In a separate bowl, whisk together the flour, baking powder, baking soda, salt, and shredded coconut. Gradually fold the dry ingredients into the wet mixture using a spatula. Mix until just combined, taking care not to overmix to maintain a soft texture.
Pour the batter into your prepared pan and smooth the top evenly. Bake for 40–45 minutes, or until a toothpick inserted into the center comes out clean. If the top begins to brown too quickly, loosely tent the pan with foil for the final 10–15 minutes.
Let the loaf rest in the pan for 10 minutes before transferring it to a wire rack. It must cool completely before you begin frosting.