Preheat oven to 350°F. Butter a 9-by-5-inch loaf pan, then line with an 18-inch-long sheet of parchment or wax paper (letting ends hang over long sides of pan). Butter parchment.
Using an electric mixer, cream butter, 1 1/2 cups sugar and lemon peel at high speed for 5 minutes. Add eggs and yolk one at a time, beating well after each addition. Beat in milk, 2 Tbsp. lemon juice, vanilla extract, baking powder and salt at medium speed. Mix in flour at low speed until batter is smooth. Scrape batter into pan, smooth top and bake until a tester comes out clean, about 1 hour.
Meanwhile, make glaze: In a small bowl, whisk remaining 1/2 cup sugar and 1/2 cup lemon juice until sugar dissolves.
Let cake cool in pan for 10 minutes, and then grasp parchment paper on sides to transfer cake to a rack. Prick top of cake all over with a toothpick. Brush one-quarter of glaze on loaf; let sit for 5 minutes. Repeat two times (reserving last quarter of glaze) and then let cake cool completely. Remove cake from paper and place it on a platter. Brush top with remaining glaze. Sift confectioners' sugar over top if desired.