Heat 2 tablespoons of the oil in a large pot over medium heat.
Add the poblano, garlic, all but ⅓ cup of the onion, ¾ teaspoon salt, and ½ teaspoon pepper.
Cook, stirring occasionally, until the vegetables are tender, 8 to 10 minutes. Stir in the cumin.
Add the beans and ½ cup water. Mash some of the beans with a fork or potato masher. Bring to a boil.
Reduce heat and simmer, stirring occasionally, until the soup is slightly thickened, 6 to 8 minutes.
For the salsa, combine the diced jalapeno, cilantro, remaining 1/3 cup onion, remaining salt and pepper, 2 tsp olive oil, and lime juice. Top soup with salsa to serve.