1. Bring broth to a simmer in a large Dutch oven or other heavy pot. Meanwhile, tear flesh of chilis into a few pieces, letting seeds fall out. Transfer to a blender. Add hot broth and cover blender; let sit until chiles are softened, 25–30 minutes. Blend on high speed until smooth, about 1 minute; set aside.
2. Heat 2 Tbsp. oil in same pot over medium-high. Season turkey all over with salt. Add half of turkey to pot and smash down with a wooden spoon to flatten against surface of pot and break into smaller clumps. Cook, undisturbed, until underside is browned, top side is no longer pink, and most of the liquid is cooked off, 8–10 minutes. Using a slotted spoon, transfer to a plate. Repeat with another 2 Tbsp. oil and remaining turkey.
3. Heat remaining 2 Tbsp. oil in same pot over medium. Add onions and garlic; season with salt and stir to combine. Cover pot and cook, stirring occasionally, until onions are translucent, very soft, and starting to brown around the edges, 10–12 minutes. Add cumin and oregano and cook, stirring frequently, until fragrant and sizzling, about 1 minute. Add tomato paste and cook, stirring to coat vegetables, until paste darkens slightly, about 1 minute. Add cider, cinnamon, and bay leaves and scrape up any brown bits stuck to bottom of pot (if you’re not using cider, skip this step and increase water in next step to 5 cups). Bring to a lively simmer and cook, stirring occasionally, until liquid is mostly reduced, about 5 minutes.
4. Add reserved chile purée, beans, and maple syrup. Season with salt. Bring to a boil, reduce heat to medium-low, and gently simmer, partially covered and stirring occasionally, until liquid is thickened and very flavorful, about 1 hour. Add turkey and simmer until flavors have melded, about 30 minutes longer.
5. Meanwhile, stir yogurt, lime juice, and cilantro in a small bowl to combine. Season to taste with salt.
6. Taste chili and season with salt and pepper if needed. Divide chili among bowls. Top with a dollop of yogurt-lime sauce and some broken tostadas.