Place the turkey carcass in a large stockpot and cover it with cold water. Add the kosher salt, black pepper, thyme, and bay leaf. Bring to a boil. Skim the foam off the surface. Reduce the heat to medium-low so the stock is at a gentle simmer. Cook, partially covered, for 3 hours.
Remove the bones and bay leaf and discard. Strain the stock through a strainer. Measure how much liquid you have - you should now have about 2 quarts of stock. If you have less, add enough water to get to 2 quarts. Wipe the stockpot clean with moist paper towels.
In the clean stockpot, heat the olive oil over medium-high heat for about 2 minutes. Add the carrots, celery, and onion. Cook, stirring occasionally, until the vegetables are tender-crisp, for about 5 minutes. You want the onions golden, not browned. If the bottom of the stockpot becomes too dry, add a splash of water. Add the turkey meat.
Skim the fat layer from the top of the broth if you wish, and then add it back to the stockpot. Bring back to a simmer and cook just until heated through, about 5 more minutes.
Serve immediately.
Video
Notes
The sodium content is an estimate.
It's up to you if you'd like to skim the fat layer off the top of the turkey stock. I prefer not to since the fat adds so much flavor. If you decide to skim it, it's easier to allow the stock to cool completely in the fridge. The fat will float to the top and harden, making it easy to remove. Then, you can reheat the stock and proceed to make the soup.
To make a creamy soup, stir half a cup of heavy cream into the strained broth, then add it to the saucepan with the sauteed vegetables and gently heat the soup for a few minutes, taking care not to bring it to a boil.
You can keep the leftovers in an airtight container in the fridge for up to four days. I reheat bowls or mugs of them in the microwave, covered, at 50% power. You can also freeze individual portions in mugs. Seal their tops with a few layers of plastic wrap before freezing.