Warm the olive oil in a large stockpot over medium heat. Sauté the diced onion until softened and translucent, about 5 minutes. Stir in the minced garlic and cook for 1 minute more.
Sprinkle in the cumin, oregano, chili powder, cayenne, salt, and black pepper. Stir the spices into the onion and garlic mixture until aromatic, about 30 seconds.
In a separate bowl, mash one cup of the white beans using a potato masher or fork until mostly smooth; this will help create a creamier consistency.
To the pot, add the chicken broth, both whole and mashed beans, green chiles, fire-roasted tomatoes, and corn. Stir thoroughly to combine.
Submerge the raw chicken breasts into the pot. Increase heat to medium-high and bring the mixture to a gentle boil. Once boiling, reduce the heat to low and let it simmer uncovered for 15 to 20 minutes, or until the chicken is thoroughly cooked and tender.
Turn off the heat and carefully remove the chicken breasts from the pot. Set them aside to cool slightly, then shred the meat using two forks.
Return the shredded chicken to the chili. Pour in the freshly squeezed lime juice and stir in the Greek yogurt until fully incorporated.
Serve hot, topped with your choice of garnishes such as cilantro, yogurt, cheese, jalapeños, lime wedges, or tortilla chips.