Preheat oven to 350°F. Spray a 9 inch loaf pan with nonstick spray and line it with parchment paper, leaving enough overhang on the long sides to help lift the loaf out after baking.
Add the granulated sugar and lemon zest to a large mixing bowl. Use your fingers to rub the zest into the sugar until the mixture is fragrant and slightly damp.
Add the butter and beat with a hand mixer until light and fluffy, about 2 to 3 minutes. Add the eggs one at a time, mixing well after each addition before adding the next.
Add the Greek yogurt, milk, lemon juice, vanilla extract, baking powder, baking soda, and salt. Mix until smooth and fully combined.
Add the flour and switch to a spatula or wooden spoon. Fold the flour into the batter just until the last dry streaks disappear. Do not overmix.
Spread the batter evenly into the prepared loaf pan and smooth the top. Bake for 50 to 55 minutes, or until the top is golden and a toothpick inserted into the center comes out clean or with a few moist crumbs.
Let the lemon loaf cool in the pan for 10 to 15 minutes, then use the parchment paper to lift it out and transfer it to a wire rack. Cool completely before adding the icing.
To make the icing, whisk the powdered sugar, milk, lemon juice, and vanilla extract together until smooth and shiny. The icing should be thick but still pourable. If it is too thick, add a small splash of milk or lemon juice. If it is too thin, whisk in a little more powdered sugar.
Pour the icing over the cooled lemon loaf and gently spread it over the top with the back of a spoon. Let the icing set before slicing. To speed up the setting time, place the loaf in the refrigerator for about 10 minutes.