Preheat oven to 350°F (180°C). Grease a 9×5-inch loaf pan and line with parchment. Beat the butter and sugar until pale and fluffy. Add eggs one at a time, then mix in lemon zest and vanilla.

In a small bowl, whisk flour, baking powder, and salt. In another bowl, stir yogurt and ¼ cup lemon juice together. With a spatula, gently fold dry ingredients and yogurt mixture into the batter in alternating additions, starting and ending with the dry ingredients. Fold just until no streaks remain.

Pour batter into the prepared pan and smooth the top. Bake 50 to 55 minutes, until a toothpick comes out with a few moist crumbs. Cool in the pan for 10 minutes, then remove to a rack and cool completely.

Whisk powdered sugar, 2 tbsp lemon juice, and melted butter for the glaze. Spoon over the cooled cake and let it set for about 20 minutes before slicing.