Heat oil in a large pot over medium-high heat. Add onions, poblano, and jalapeño, cooking until onions are soft and translucent. This step builds a flavorful base for your soup.
Pour in the broth, enchilada sauce, diced tomatoes, beans, corn, and brown rice. Sprinkle in smoked paprika, garlic powder, cumin, and cayenne. Stir to combine so flavors meld together.
For stovetop, bring to a simmer, reduce heat, cover, and cook 25–30 minutes until rice is tender. For Instant Pot, pressure cook 15 minutes and allow 5-minute natural release.
Turn off heat and fold in shredded cheese. Let it melt fully to make the soup creamy and rich. Plant-based cheese works well for dairy-free versions.
Ladle soup into bowls and top with cilantro, sour cream, lime wedges, and tortilla chips. Each garnish adds extra flavor and texture for a satisfying meal.