Place the turkey carcass, carrots, celery stalks, onion and parsley in a large soup pot with at least 2 ½ quarts of water, or enough water to cover by about 2 inches. Bring to a boil, season with salt and pepper, reduce heat, and simmer,covered, for about two hours, until the aroma of the stock permeates the kitchen (!). Remove from heat and carefully strain out the solids by pouring the hot stock through a colander into a large bowl or another pot.

Let the solids cool. Clean off whatever meat is salvageable from the carcass, and collect whatever meat may have fallen off the bone while simmering the carcass. Set this aside until ready to use.

Measure out the stock. You should have at least 8 cups of liquid. Using that same pot, warm the olive oil over medium heat and cook the leek until soft and glistening. Stir in the garlic and turkey shreds. Pour the strained turkey stock into the pot. Bring to a simmer and add the rice. Cook over gentle heat until the rice is tender, about a half hour. Season with salt and pepper as needed.

Toward the end of the simmering, whisk the egg yolks and egg in a medium bowl until thick and frothy. Slowly add the lemon juice, whisking all the while, until the mixture turns pale yellow and creamy.

Turn off the heat on the soup. Take a ladleful at a time, avoiding to get any solids into the ladle, and very slowly drizzle the hot soup broth into the egg-lemon mixture, whisking vigorously. Repeat this 4 to 5 more times. Pour the avgolemono into the pot and swirl it around to mix thoroughly. Serve immediately.