Preheat oven to 350°. Grease a 9" x 5" loaf pan with cooking spray, then line with parchment, leaving a 2" overhang on 2 longer sides. In a medium bowl, whisk flour, baking powder, and salt.

In a large bowl or the large bowl of a stand mixer, combine 1 cup granulated sugar and 2 tablespoons plus 1 ½ teaspoons Key lime zest. Using your fingers, rub sugar and zest together until sugar is slightly damp and very fragrant. Add butter to bowl. Using a handheld mixer on medium speed (or using the stand mixer fitted with the paddle attachment), beat butter and sugar mixture until light and fluffy, about 2 minutes. Add eggs, one at a time, beating to blend after each addition. Beat in vanilla. Add dry ingredients and beat on low speed until just a few dry streaks remain. Beat in sour cream until just combined. Pour batter into prepared pan; smooth top.

Bake cake until a tester inserted into the center comes out clean, 55 to 60 minutes.

Meanwhile, in a small saucepan over medium-high heat, bring ¼ cup Key lime juice and remaining ¼ cup granulated sugar to a boil, stirring constantly. Cook, stirring, until sugar is dissolved, about 2 minutes. Remove from heat.

Poke cake all over with a long skewer or thin knife. Pour Key lime syrup over. Transfer to a wire rack and let cool completely.

Using parchment overhang, remove cake from pan and transfer to a wire rack set in a baking sheet. Remove parchment. In a small bowl, whisk confectioners’ sugar, a pinch of salt, and remaining 2 tablespoons lime juice and 1 ½ teaspoons lime zest until smooth. Drizzle over cake and let stand until glaze is set, about 10 minutes.