Prepare the pressure canner, jars, and lids. Add water to the canner following your manufacturer's directions, and keep the clean jars hot in it.
Peel and slice the carrots, dice the celery and onions, and mince the garlic. Cut or shred the roasted turkey into bite-size pieces, leaving out any skin, bone, or gristle.
Put the turkey, vegetables, and garlic in a large stockpot, and pour in the turkey stock, chicken broth, or water, adding a little more if needed to cover the solids. Stir in the poultry seasoning, salt, and pepper if you're using them.
Bring the soup to a boil, and boil it for 5 minutes. The vegetables finish cooking in the jars, so they don't need to be tender yet. Taste the broth and adjust the salt.
Use a slotted spoon to pack each hot jar with turkey and vegetables to 1 inch headspace, without pressing down hard.
Ladle hot broth over the solids to fill the gaps, leaving 1 inch headspace.
Remove air bubbles, top up with hot broth if a jar came up short, wipe the rims clean, and apply the lids and bands fingertip tight.
Load the jars into the canner, lock the lid, and vent a steady stream of steam for 10 minutes before you bring it to pressure.
Process at 10 lbs pressure in a weighted gauge canner or 11 lbs in a dial gauge canner, pints for 75 minutes or quarts for 90 minutes, adjusting for altitude (see notes).
Turn off the heat and let the canner return to zero pressure on its own. Wait 5 minutes, remove the lid, and leave the jars in the canner another 10 minutes.
Lift the jars out and let them cool undisturbed for 12 to 24 hours. Check the seals, remove the bands, label the jars, and store them in a cool, dark place.