In a medium bowl, whisk together the flour, baking powder, and salt.
Heat your oven to 350°F. Grease an 8 1/2 by 4 1/4 -inch loaf pan. Line the bottom and two long sides of the pan with parchment paper.
In a larger bowl, combine 1 cup of the granulated sugar and the zest. Rub them together with your fingertips to extract as much grapefruit flavor as possible.
Add yogurt, eggs, and vanilla to the zest-sugar mixture. Whisk to combine.
Whisk the flour mixture into the yogurt mixture, then whisk in the oil.
Pour the batter into the prepared pan and smooth it out. Bake for about 50 minutes, or until a cake tester inserted into the center comes out clean.
While the cake bakes, combine 1/3 cup grapefruit juice with 1 tablespoon granulated sugar in a small saucepan. Stir over low heat until the sugar dissolves.
When the cake comes out of the oven, transfer it to a cooling rack. Pour or brush the grapefruit-sugar mixture over the whole cake. Poke holes in the cake with a toothpick to help it absorb. Let the cake cool completely.
Combine the powdered sugar and 2 tablespoons grapefruit juice in a bowl. Drizzle it over the cake.
Option: To one-bowl the cake, combine grapefruit zest and sugar in a large bowl and rub them together. Add yogurt, eggs, oil, and vanilla. Whisk until mixed. Sprinkle the surface with salt and baking powder and whisk until combined. Add flour and mix until it disappears. Continue with baking and glazing instructions above.