- Whisk together pomegranate juice, honey, vinegar, and cardamom in medium saucepan.
- Peel and quarter quince; cut out fuzzy core and seeds. Cut each quarter into 4 or 5 wedges and add to pomegranate-juice mixture; bring to boil. Reduce heat to low and cook, covered, until quince is tender, about 12 minutes. (If using apple, cook until softened, about 5 minutes.) Remove pan from heat; with slotted spoon, transfer quince to small bowl. Reserve poaching liquid.
- To make vinaigrette, add oil, garlic, salt, and pepper to poaching liquid; bring to boil and cook until reduced to about 1/3 cup, about 8 minutes. Discard garlic. Let cool to room temperature.
- Toss together arugula, quince, pecans, pomegranate seeds, and vinaigrette in large salad bowl until coated evenly with vinaigrette. Divide salad evenly among 6 plates and top with cheese shavings.
- Per serving: about 1 1/2 cups
Quinces are greenish-yellow and resemble lumpy pears. Although available year-round, California-grown quinces are usually in stores in the autumn. Choose those that are free of blemishes and discolorations. When ripe, they turn pale yellow and remain quite hard but exude a delicate, sweet floral aroma.