Preheat the oven to 180°C. Butter and flour a loaf pan.
In a large bowl, cream together coconut oil and sugar. Beat in the eggs, one at a time. Add the greek yogurt, lemon zest, and juice and mix well.
In a small bowl, combine flour, salt, and baking soda. Gradually add the dry ingredients to the batter and mix until just combined. Pour the batter into the prepared loaf pan and bake for 45-50 minutes, until a tooth pick inserted comes out clean. Let the cake cool in the pan for 10 minutes before transferring it to a wire rack.
For the icing, whisk together the icing sugar, lemon juice and just enough milk for a very thick consistency. Pour the icing over the cake, letting it drip over the sides. Wait for the icing to harden before cutting the cake.