Heat the olive oil in a large pot over medium heat. Add the onion and cook for 5 minutes until soft.
Stir in the garlic, smoked paprika, red pepper flakes and black pepper. Cook for 1 minute until fragrant.
Add the smoked turkey, broth and water. Bring to a boil, then lower to a simmer for 15 minutes.
Add the collard greens a few handfuls at a time, stirring until they wilt down.
Cover and simmer for 60 minutes, stirring now and then, until the greens are silky and tender.
Pull the meat off the turkey, shred it and stir it back in. Add the vinegar, taste and adjust the salt. Serve with the pot likker.