Preheat the oven to 160°C (320°F). Line a 9.8" loaf pan with baking paper.
Cream the butter, sugar, and lemon zest: Add the softened butter, sugar, and lemon zest to a bowl. Mix well using a hand mixer.
Add the oil: Once combined, add the avocado oil and mix briefly.
Scrape the bowl: Scrape down the sides and base of the bowl to ensure everything is well mixed.
Add the eggs: Add the eggs one at a time, mixing well between each addition.
Add vanilla and lemon juice: Add the vanilla extract and mix briefly. Then, add the lemon juice and mix briefly. The mixture will curdle, but that’s fine.
Add the milk: Pour in the milk and mix until well combined.
Sift the dry ingredients: Place a strainer over the bowl and sift in the flour and baking powder.
Fold the batter: Use a silicone spatula to fold the flour into the wet mixture. Mix briefly until no visible flour remains. Avoid overmixing.
Bake: Transfer the batter into the prepared loaf pan. Bake for 55-65 minutes, or until a toothpick inserted into the center comes out clean.
Cool: Let the cake cool in the pan for 10 minutes, then transfer it to a wire rack to cool completely.
Make the frosting: Combine powdered sugar, lemon juice, and milk in a bowl. Whisk until smooth and pour over the cooled cake.
Serve: Optionally garnish with ground pistachios. Slice and enjoy your fresh lemon loaf cake.
