Make stock: Peel and quarter 3 onions. Cut 3 celery stalks and 3 carrots into 1-in. lengths. Cut leeks (white and light green parts only) into 3/4-in. lengths, rinse thoroughly, and drain. Cut rosemary sprig in half. Divide cut-up vegetables, rosemary, thyme sprigs, parsley sprigs, bay leaves, 1 tsp. salt, the peppercorns, and turkey carcass between two 8-qt. pots (or use 1 very large pot). Add 20 cups cold water to each pot and bring almost to a boil over high heat. Just before stock reaches a boil, lower heat. Maintain a very gentle simmer for 4 hours, skimming off foam for the first 30 minutes.
Strain stock through a colander into a large bowl. Wash pot and line colander with cheesecloth. Strain stock through cheesecloth-lined colander back into pot.
Chill stock, covered, at least 5 hours or overnight. Skim fat from stock and discard. You should have at least 15 cups stock (if you don't, add enough water to make up the difference).
Make soup: Cut remaining 2 onions and remaining 4 celery stalks into 1/4-in. dice. Peel remaining 4 carrots, halve lengthwise, and slice 1/4 in. thick.
Bring 15 cups stock (freeze any leftover for another use) to a simmer over medium-high heat.
Heat extra-virgin olive oil in an 8-qt. pot over medium heat. Add diced onions, celery, and carrots, plus 1/2 tsp. each salt and pepper. Cook until vegetables are softened but not browned, about 20 minutes.
Stir in garlic and chopped thyme and cook, stirring occasionally, 1 to 2 minutes, or until garlic is fragrant. Add hot stock and bring to a simmer. Stir in remaining 3/4 tsp. salt and 1/2 tsp. pepper.
Add turkey meat and beans to soup and simmer until heated through, 2 to 3 minutes. Stir in chopped parsley. Add salt to taste. Serve soup in shallow bowls drizzled with olio nuovo and sprinkled with parmesan.
Make ahead: Up to 3 months, frozen.
Types of Olive Oil
The USDA recently adopted these international standards for olive oil grades.
Extra-Virgin. The highest grade you can buy; the olives are pressed without using chemicals or adding heat, and they go through the press only once. To qualify as extra-virgin, the oil must also be free of specified taste "defects" and have less than 5 percent acidity.
Olio Nuovo. Italian for "new oil," this is an extra-virgin oil less than 3 months old. Intensely green, with a pungent, vibrant taste, it quickly loses its bite, so use it right away. (Most California-produced brands of olio nuovo appear in late fall, soon after the harvest, and often sell out quickly.)
Virgin. A lower grade than extra-virgin, lighter in flavor and higher in acidity.
Olive oil. Virgin oil blended, in varying ratios, with refined oil, which comes from olives that have already been pressed at least once; heat or chemicals may be used to extract more oil from the paste. Neutral in flavor and cheap.
Light. Refined oil with a small amount of virgin oil, if any; may also have other vegetable oils. "Light" doesn't mean lower in calories, just a lighter taste.
Infused/Flavored. Oil in which fruit or herbs were steeped. Can also mean the olives were pressed with fruit or herbs, or that fruit or herb oils were added to the oil after pressing; both methods result in a more intense flavor than the steeping.
Note: Nutritional analysis is per serving.