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 that the stock is at a gentle simmer. Cook, partially covered, for 3 hours.
Remove the bones and the bay leaf and discard. Strain the stock through a strainer. Measure how much liquid you have - you should now have about 2 quarts of soup. 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, about 2 minutes. Add the carrots, celery, and onion. Cook, stirring occasionally, until the vegetables are tender-crisp. You want the onions golden, not browned. If the bottom of the stockpot becomes too dry, add a splash of water.
Skim the fat layer from the top of the broth if you wish, then add it back to the stockpot. Add the turkey meat. Bring back to a simmer and cook just until heated through, about 5 more minutes.
The sodium content is an estimate. If this is of concern to you, please verify it independently.It's up to you if you'd like to skim the fat layer off the top of the soup. I actually prefer not to, since the fat adds so much flavor. If you decide to skim it, it's easier to allow the broth to cool completely in the fridge. The fat will float to the top and harden, and it will be very easy to remove. Then you can reheat the broth and proceed to make the soup.