Heat the olive oil in a large pot over medium heat, then add the diced onion and cook until translucent and fragrant, about 5 minutes.
Add the minced garlic and diced poblano peppers to the pot, cooking for another 3-4 minutes until the peppers soften and the mixture smells spicy and smoky.
Sprinkle in the ground cumin, stirring to evenly coat the vegetables, and cook for 1 minute to toast the spices.
Pour in the chicken broth and bring the mixture to a gentle simmer, allowing the flavors to meld for 5 minutes.
Add the shredded chicken and drained white beans to the pot, stirring well to combine.
Pour in the milk and stir gently, then heat everything until the chili is bubbling softly, about 5 minutes.
Gradually stir in the shredded Monterey Jack cheese, letting it melt slowly while stirring over low heat for a creamy, cohesive texture.
Squeeze fresh lime juice into the chili and season with salt and pepper to taste, stirring again to incorporate bright, tangy flavors.
Cook for another 2-3 minutes to ensure everything is heated through and flavors are well combined.
Serve hot in bowls, garnished with extra cheese, a squeeze of lime, or fresh herbs if desired. Enjoy the hearty, smoky, and tangy chili right away!