Preheat oven to 350°F (175°C). Line a 9x5-inch loaf pan with parchment paper.
Rub lemon zest into sugar until it looks like wet sand.
Cream the softened butter and lemon-sugar mixture until light and fluffy.
Beat in eggs one at a time until smooth.
In a separate bowl, whisk flour, baking powder, and salt.
Alternate adding dry ingredients and buttermilk mixture to the butter, starting and ending with dry. Mix just until combined.
Pour batter into the pan and bake for 50-55 minutes.
Make soak syrup: heat 3 tbsp lemon juice and 1 tbsp sugar until dissolved. Brush over hot loaf.
Cool in pan 15 minutes, then cool completely on a rack.
Whisk powdered sugar, 2 tbsp lemon juice, and 1 tbsp milk until smooth. Pour over cooled loaf.
Let icing set for 10 minutes before slicing.