Remove the smoked turkey wings from the pot and let cool. Once cooled, pull the meat from the bones, discarding the skin, bones, and cartilage. Chop or shred the turkey meat into bite-sized pieces.
Return the shredded turkey to the pot. Add salt, starting with 1 teaspoon, and stir well. Taste and add more salt as needed. Season with freshly ground black pepper to taste.
Stir in 1 tablespoon of apple cider vinegar. Let the peas simmer for an additional 10 minutes to allow the turkey to warm through and flavors to meld.
Remove from heat and let the pot rest, covered, for 10 minutes. The flavors will continue to develop, and the broth will thicken slightly as it cools.