Preheat your oven to 400°F (200°C). Place the whole poblano peppers on a baking sheet and roast for about 20 minutes until the skins are blistered and charred.
Remove the roasted peppers from the oven, cover with a kitchen towel, and let them steam for 10 minutes. After they cool, peel off the skins, remove seeds, and chop the peppers.
In a large pot, heat olive oil over medium heat. Add the diced onion and sauté for about 5 minutes until translucent. Then, add the minced garlic and sauté for an additional 1 minute.
Place the chicken breasts into the pot, and season with salt, pepper, cumin, and smoked paprika. Pour in the chicken broth, and bring to a boil. Reduce heat and let it simmer for about 15 minutes until the chicken is cooked through.
Remove the chicken from the pot and shred it using two forks. Return the shredded chicken to the pot along with the chopped roasted poblano peppers and stir to combine.
Pour in the heavy cream and stir until well mixed. Allow the soup to cook for an additional 5 minutes on low heat.
For a smoother texture, blend part of the soup using an immersion blender or traditional blender. Taste the soup and adjust the seasoning with salt and pepper if necessary.
Serve hot, garnished with fresh cilantro.