Trim silver skin from the venison and cut it into 1.5- to 2-inch chunks. Pat dry and season with the salt and black pepper.

Toast the stemmed, seeded guajillo and ancho chiles in a dry skillet over medium heat for about 10-15 seconds per side, just until fragrant. Do not burn. Transfer to a bowl, cover with hot water, and soak for 15 minutes; drain.

Blend the softened chiles with the tomatoes, quartered onion, garlic, apple cider vinegar, cumin, oregano, cinnamon, ground cloves, and 1 cup of the stock until smooth. Strain through a fine-mesh sieve if a silkier broth is desired.

Heat the oil in a large Dutch oven over medium-high heat. Sear the venison in batches for about 6-8 minutes total, turning to brown several sides without crowding the pot. Transfer to a plate.

Reduce heat to medium. Pour the blended chile mixture into the pot carefully and cook, stirring, for 3-4 minutes, scraping up the browned bits.

Return the venison and its juices to the pot. Add the remaining 3 cups stock and bay leaves. Bring just to a simmer, then cover and reduce heat to low.

Gently simmer for about 3-3.5 hours, checking occasionally, until the venison is easily pierced and pulled apart with a fork. Add a splash of stock or water if the liquid gets too low; avoid a hard boil.

Remove the bay leaves. Shred some of the venison with two forks, leaving some pieces intact. Stir into the broth, taste, and adjust salt or vinegar as desired. Skim excess surface fat if needed.

Ladle into six bowls and top with diced white onion, cilantro, and lime juice. Serve with warm corn tortillas, if desired.