Slice a small sliver from the top and bottom of grapefruit to reveal the flesh. Set the grapefruit in a small saucepan and cover with water. Bring to a boil, and cook for 1 minute. Drain. Repeat these steps two more times (covering with water, boiling, cooking for 1 minute, and draining).

Place the blanched grapefruit in a medium pot with 1 cup of the sugar and 3 cups of water. Cook over medium heat for 20-30 minutes, until you can easily pierce the grapefruit rind with a fork and the sugar has dissolved. Use tongs to remove the grapefruit from the pot, and place on a plate to cool. Reserve the sugar liquid, setting it aside to cool.

Preheat oven to 350℉. Grease and flour a 10-inch round cake pan, set aside.

In a mixing bowl, whisk together the flour, baking powder, baking soda, salt, and chopped rosemary. Set aside.

Slice the cooled grapefruit in half and scoop the flesh into the bowl of a food processor. Scrape out most of the pith from the rind, and discard. Slice the rind into large chunks and add to the food processor. Add the eggs, and pulse to combine, until you have a chunky purée. Add the olive oil and the last of the sugar, and pulse to incorporate. In batches, add the flour mixture, pulsing after each addition to combine. Don’t overmix.

Transfer the batter into the prepared cake pan. Bake 40-50 minutes, until the top and edges begin to turn golden brown, and a toothpick inserted into the center comes out clean. Set on a wire rack. Use a pastry brush to brush some of the reserved grapefruit-sugar liquid on top of the cake. Let cool on the rack for 15 minutes, then remove from pan and cool completely.

When cool, dust with powdered sugar, and slice and serve.