Preheat the oven to 325°F.
In a baking dish, coat the chicken in the Cajun seasoning and salt. Drizzle with olive oil, cover with aluminum foil and roast for 75 to 90 minutes, or until the chicken registers 165°F internal temperature and is very tender.
In a small saucepan, make a roux. Melt the butter on low. When melted, whisk in the flour and keep whisking every 2 minutes until a nice blonde color is achieved, about 12-15 minutes.
Meanwhile, in a separate large pot, heat the oil until smoking. Add the onion, celery, and carrots and cook until translucent. Add the andouille sausage and poblano pepper and sauté until the pepper is soft and the sausage is rendered.
Add the garlic and Cajun seasoning and cook until fragrant, 1-2 minutes. Add the chicken stock, 1/2 teaspoon salt, tomatoes and tomato paste. Stir to combine. Slowly add in the roux, one tablespoon at a time, stirring constantly to make sure each addition is completely dissolved before adding the next. Bring to a boil to thicken, then lower to a simmer and cook for 5-10 minutes.
Add the sherry vinegar, season with salt to taste and cook for another 2 minutes. Stir in the beans.
In a blender, combine all of the chimi-verde ingredients except the oil until smooth. With the motor running, gradually drizzle in the oil until the sauce is emulsified. Season to taste.
Divide the Cajun beans between bowls and top with chicken legs and chimi-verde sauce.