lemon 🍋 cake

Cake
1 ½ cups all-purpose flour
1 ½ tsp baking powder
1 Tbsp lemon zest
½ tsp salt
½ cup unsalted butter, room temperature
1 cup granulated sugar
2 large eggs, room temperature
1 tsp vanilla extract
2 Tbsp lemon juice, about 1/2 lemon
½ cup buttermilk, see below for substitution
For the lemon syrup:
¼ cup lemon juice, about 1 lemon
3 Tbsp powdered sugar
For the lemon icing:
1 cup powdered sugar, sifted
1 ½ Tbsp lemon juice
1 Tbsp milk