Using a box grater, coarsely shred the mushrooms. In a large, deep, heavy skillet, heat 1 tablespoon olive oil over medium heat. Add the mushrooms and cook, stirring occasionally, until browned, about 10 minutes. Season with salt and pepper; transfer to a plate.
Add the remaining 1 tablespoon olive oil and the onion to the skillet over medium heat and cook, stirring, until golden, about 5 minutes. Add the chili powder and cumin and cook, stirring, for 1 minute. Add the ground bison meat and cook, breaking up the meat, until no longer pink, about 3 minutes.
Stir the tomatoes, black beans and chickpeas into the chili. Partially cover and simmer until thickened, about 15 minutes.