Place the turkey carcass in a large stockpot and cover 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. 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.