Remove the flesh of each grapefruit quarter, leaving pith and peel; reserve flesh for another use. Slice peels into ¼"-wide strips; transfer to a 4-qt. saucepan. Cover with water by 1”; bring to a boil and cook for 1 minute. Strain and repeat process 3 more times; set peels aside.
Add 1 cup sugar, cardamom, cloves, cinnamon, and 1 cup water to pan; bring to a boil. Reduce heat to low and add peels; cook, stirring occasionally, until peels are very soft, about 1 hour. Let peels cool in syrup, then strain, discarding spices; reserve syrup for another use. Spread peels evenly on a wire rack set over a baking sheet; let sit until dried, 2–3 hours. Toss peels with remaining sugar; store in an airtight container between layers of wax paper up to 1 week.