Heat 1 Tbsp. vegetable oil in a heavy large pot over medium heat.
Add 2 medium yellow onions, chopped, and 1–2 medium jalapeños, seeded, finely chopped; cook, stirring often, until lightly browned in spots and tender, about 10 minutes.
Add 1½ tsp. dried oregano and 1½ tsp. ground cumin and cook, stirring often, until fragrant, about 1 minute.
Increase burner to medium-high heat.
Add 1½ lb. ground turkey and cook, stirring and breaking turkey up with a spoon until no longer pink, 5–8 minutes.
Stir in ¼ cup chili powder, 2 bay leaves, 1 Tbsp. unsweetened cocoa powder, 1½ tsp. Diamond Crystal or ¾ tsp. Morton kosher salt, and ¼ tsp. ground cinnamon and cook, stirring often, until spices are fragrant, about 3 minutes.
Add three 15-oz. cans small white beans, such as cannellini, rinsed, drained, 3 cups low-sodium chicken broth, and one 28-oz. can crushed tomatoes.
Bring to a boil.
Reduce heat; simmer, stirring occasionally, until thickened slightly and flavors have melded, 45–50 minutes.
Gently crush some beans using the back of a spatula until chili is as thick as you would like.
Stir in juice of 1 medium lime, taste, and season with salt, as needed.
Discard bay leaves.
Ladle chili into bowls.
Top with grated Cotija cheese, sliced red onion, chopped cilantro, and plain yogurt or sour cream, if using.
Serve with lime wedges.
