Heat 1 tbsp of the olive oil in a 26 cm / 5.5-quart Dutch oven over medium-high heat. Add half the venison and break it into crumbles with a wooden spoon. Cook for 4 to 5 minutes until no pink remains and it reads 160 F / 71 C, then move it to a bowl. Brown the second half the same way.
Lower the heat to medium. Add the remaining 1 tbsp oil, the onion and the bell pepper, and cook for 5 minutes until the onion is soft and translucent. Stir in the garlic and tomato paste and cook for 1 minute.
Add the chili powder, cumin, smoked paprika, oregano, cocoa, salt and pepper. Stir for 1 minute until fragrant and the paste darkens, then return the venison and its juices to the pot.
Pour in the crushed tomatoes and broth, scraping up the browned bits from the bottom. Bring to a boil over high heat, about 2 minutes, then drop the heat to low so it barely bubbles. Partly cover and simmer for 30 minutes, stirring every 10 minutes.
Stir in the black beans and kidney beans and simmer partly covered for 15 minutes more, until a spoon dragged across the bottom leaves a clear trail. If it looks thin, mash about 1/2 cup of the beans against the side of the pot.
Stir in the apple cider vinegar, then taste and add more salt if needed. Ladle into bowls and serve hot.