Start by preheating the oven to 350 degrees F. Grease an 8-inch x 4-inch loaf pan. If you have a sticky pan, grease it, then lay parchment in the bottom, then grease and flour.
In a large bowl, sift together the flour, baking powder, and salt. In another bowl, add the sugar. Then grate the grapefruit zest directly into the sugar. Use your hands to massage the zest into the sugar until the sugar is very fragrant and slightly wet from the oil in the zest.
Whisk the yogurt, eggs, and vanilla into the sugar and zest. Slowly whisk the dry mixture into the wet mixture.
Then use a silicone spatula to fold the grapeseed or other neutral oil into the batter, making sure the oil is fully incorporated.
Pour the batter into the prepared loaf pan and bake for 50-60 minutes. Start checking for doneness at 45-50 minutes. Cake is done when a cake tester placed in the center, comes out clean.
Grapefruit Soak
Meanwhile, in a small pan, cook the 1/3 cup grapefruit juice and the remaining 1 tablespoon sugar until the sugar is completely dissolved. Remove from heat and set aside.
Let the cake cool in the pan for 10 minutes when it comes out of the oven.
Then, while the cake is still warm, drizzle the sugar grapefruit mixture over the cake and let it soak in. Let the cake cool for about 30 minutes.
Grapefruit Glaze
Meanwhile, whisk together 1 cup of confectioners' sugar and 2 tablespoons of grapefruit juice for the glaze.
After about 30 minutes, carefully invert the cake onto a plate, then invert again onto a wire rack. Set the wire rack over a sheet pan or over a sheet of plastic wrap for easy cleanup. Let the cake cool longer if it feels warm to the touch. When the cake is cool, pour the glaze over the top, letting it drip down the sides. Let the glaze set, then transfer it to a serving plate.