Heat the olive oil in your large pot over medium heat. Add the diced onion and cook for about 4 minutes, stirring occasionally, until it turns soft and slightly golden. Add the minced garlic during the last minute so it doesn't burn. This step builds the flavor foundation for the entire chili, so don't rush past it.
Stir in the cumin, oregano, and chili powder, letting them toast in the oil for about 30 seconds until fragrant. Add the canned green chilies right after so the spices coat them evenly. This quick toasting step wakes up the spices and keeps the chili from tasting flat or dusty.
Add the chicken broth and drained white beans to the pot, stirring to combine everything. Bring the mixture up to a gentle boil, then reduce the heat so it simmers steadily. Let it cook uncovered for about 10 minutes so the beans soften slightly and the flavors start to come together.
Add your shredded leftover chicken directly into the simmering pot. Since the chicken is already cooked, you're just warming it through and letting it soak up the broth's flavor. Stir occasionally and let it simmer for another 5 to 7 minutes.
Lower the heat slightly, then add the softened cream cheese in small chunks, stirring until it fully melts into the broth. Follow with the shredded Monterey Jack or Velveeta, stirring continuously so it melts smoothly instead of clumping. This is the step that turns a basic soup into a rich, creamy chili.
Turn off the heat and squeeze in the fresh lime juice, then taste and adjust with salt and pepper. Give it one final stir so everything is evenly distributed. Let the chili sit for a couple of minutes before ladling it into bowls, since it thickens slightly as it rests.