Preparation

• Line a 10×5-inch metal loaf pan with aluminum foil, allowing plenty of overhang on the long sides. Use 1 Tbs. of the butter to grease the inside of the pan, paying special attention to the short (unlined) sides.
• Position a rack in the center of the oven and heat the oven to 350°F.
• Sift the flour, baking powder, salt, and baking soda into a large bowl. Using a fork, stir in the coconut and zest, evenly distributing any clumps.
• In the bowl of a stand mixer fitted with the whisk attachment, or using a hand mixer, whisk the remaining 10 Tbs. butter and 1 cup of the sugar on medium speed until pale, scraping down the bowl occasionally with a silicone spatula, about 5 minutes. Reduce the speed to low and whisk in the eggs one at a time. Whisk in 1 Tbs. of the Key lime juice. (The batter may appear slightly curdled.) Add half of the flour mixture and mix briefly on low speed until incorporated. Add the yogurt and mix briefly on low speed until incorporated. Add the remaining flour mixture and mix on medium speed until the batter is smooth, about 1 minute.
• Scrape into the prepared pan and bake until a skewer inserted in the center comes out clean, 55 to 65 minutes.
• Meanwhile, combine the remaining 1/4 cup sugar with 1/4 cup water in a small saucepan. Bring to a boil over medium heat, stirring constantly, until the sugar dissolves. Boil one minute. Remove from the heat and add the remaining 2 Tbs. Key lime juice. Cool until needed.
• Place the cake pan on a rack. Poke all over with a cake tester or toothpick. Brush liberally with all of the lime syrup, pausing occasionally to let it soak in. Let cool for 15 minutes. Lift out by the foil and finish cooling completely before removing from the foil and slicing.