In a small bowl, combine the salt, black pepper, paprika, and cumin. Mix well. Add the seasoning to all sides of the chicken breasts.
Add 2 tablespoons olive oil to a large soup pot over medium high heat. Let heat up for 1 minute, until hot and glistening.
Add the chicken to sear 3-4 minutes on each side, until browned.
In the meantime, char the poblanos by either cooking directly on the gas stovetop racks for 2-3 minutes on each side, or by broiling in the oven 1-3 minutes on each side. Slice out the stem and seeds, then chop into small bite-sized pieces.
Add the minced garlic to the soup pot on the outsides of the chicken. Use a wooden spoon to saute it for 30 seconds.
Add the chicken broth and chopped poblanos. Bring to a boil, then cover to simmer for 20-30 minutes.
Remove the chicken with tongs to a plate. Let rest 5 minutes then shred into larger pieces between two forks. For a quicker method, use a stand mixer fitted with the whisk or paddle attachment. Add the shredded chicken back into the soup pot.
Stir in the drained white beans, lemon juice, chopped cilantro, and remaining 2 tablespoons olive oil.
