In a large soup pot over medium heat, melt butter. Add chopped onion, jalapeño, and garlic. Cook for 3–4 minutes until the onion is translucent and fragrant.

Stir in the flour to coat the veggies. Cook for about 15 minutes—stir constantly to avoid burning. The roux should thicken and lose its raw flavor.

Gradually whisk in chicken broth to prevent lumps. Add chopped roasted poblanos and remaining broth. Bring gently to a simmer.

Stir in half‑and‑half and softened cream cheese. Simmer on low, stirring frequently until the cheese melts and the base is smooth.

Fold in shredded chicken and continue to heat until the soup is bubbly and the chicken is warmed through, about 5 minutes.

Season to taste with salt and freshly ground pepper. Serve hot, garnished with cheese, cilantro, or a squeeze of lime if desired.