Heat olive oil in Dutch oven for medium-high heat until shimmering. Add onions, peppers, and a pinch of salt. Stir occasionally until softened.
Add garlic, all of the spices, and oregano. Stir and allow to bloom for 1 minute.
(If using Ground Turkey) Add ground turkey, gently breaking up with a wooden spoon. Cook until the ground turkey loses its pink color, approximately 3 to 4 minutes.
Add broth, corn, chilis, and beans. Bring to a boil and add Gaya cilantro soup base.
(If using Roasted Shredded Turkey) Add shredded turkey to Dutch oven.
Add pickled peppers, salt, and pepper to taste. Simmer covered for 6 to 10 minutes.
Add crème fraîche and lime juice. Stir and adjust seasonings if necessary.
Serve with fresh cilantro, chopped red onion, and shredded pepper jack cheese.