Preheat oven to 400°F. Line a baking sheet with foil and arrange peppers on top. Roast for 30-40 minutes, turning halfway through, until peppers are soft and skins are blackened.
Place peppers in a bowl and cover with plastic wrap. Let cool until easily handled. Then, peel off skins and discard. Slice open, lay flat, and remove stems and seeds. Stack peppers and dice.
To make the rice dish
Set a large pot over medium heat. Add olive oil and onion and sauté for 5 minutes, until translucent. Add garlic and sauté for 30 seconds more, until fragrant.
Add diced chicken and cook for 5-8 minutes.
Combine spices in a small bowl and sprinkle over chicken. Stir well to coat, then add rice and stir again. Cook 5 minutes, until rice is gently toasted.
Stir in a small amount of broth, scraping bottom of pot to release any stuck bits. Then stir in remaining broth, cream, and poblano peppers. Bring to a boil, then reduce heat to low and cover. Simmer for 15-18 minutes, stirring occasionally, until rice is tender.
Notes
Poblano peppers can be roasted 1-2 days in advance and stored in the refrigerator to reduce cooking time the day of.