Preheat your oven to 165°C (350°F). Grease and line a 9x5 inch (23x13 cm) loaf pan with parchment paper.
In a large mixing bowl, cream together the softened butter and caster sugar until light and fluffy. This should take about 3-5 minutes.
Mix in the lemon zest. Then, add the eggs one at a time, beating well after each addition to ensure they are fully incorporated.
In a separate bowl, sift together the self-raising flour, bicarbonate of soda, and salt.
Gradually add the dry ingredients to the wet mixture, mixing gently until just combined. Avoid overmixing. Add the milk and 1 tsp lemon juice, and fold into the batter until smooth.
Pour the batter into the prepared loaf pan and smooth the top with a spatula. Bake in the preheated oven for 45-55 minutes, or until a skewer inserted into the center of the cake comes out clean.
While the cake is baking, mix the 58g caster sugar and 2 tbsp lemon juice in a small bowl until the sugar dissolves.
When the cake is done, remove it from the oven and allow it to cool in the pan for about 10 minutes. Transfer the cake to a wire rack. While it's still warm, poke small holes all over the top with a skewer or fork and pour the lemon drizzle over, letting it soak in.
For the optional lemon icing, mix 35g icing sugar with 2 tsp lemon juice until smooth. Once the cake is completely cool, drizzle the icing over the top for an extra lemony finish.
Slice the cake and enjoy your delicious lemon drizzle loaf cake!