Heat the olive oil in a large heavy pot over medium heat until shimmering, 1 to 2 minutes; add the yellow onion, red bell pepper, and jalapeño pepper and cook, stirring, until softened, 5 to 7 minutes, then stir in the garlic and cook until fragrant, about 1 minute.
Add the ground turkey, increase heat to medium-high, and cook, breaking it up, until no longer pink and most moisture has evaporated, 6 to 8 minutes; season with the kosher salt and black pepper.
Stir in the chili powder, ground cumin, dried oregano, and smoked paprika and cook to bloom the spices, about 1 minute.
Add the tomato paste and cook, stirring, until it darkens slightly and coats the meat and vegetables, 1 to 2 minutes.
Pour in the canned diced tomatoes and chicken broth and bring to a lively simmer, 3 to 5 minutes, scraping up any browned bits.
Stir in the kidney beans and black beans, reduce heat to maintain a gentle simmer, and cook partially covered, stirring occasionally, 30 to 40 minutes, until the chili is thick and glossy and a spoon dragged across the surface leaves a brief trail.
Remove from heat and let rest 5 minutes; taste and adjust seasoning if needed.
Serve hot with cilantro leaves, lime wedges, sour cream, sharp cheddar cheese, and scallions as desired.