Set your oven’s broiler to high and begin preheating. Place the poblanos and corn on a large baking sheet. Rub the corn all over with a little olive oil.
Place the sheet pan under the broiler and cook until nicely charred, rotating the corn every few minutes and flipping the poblanos as needed so they char on both sides.
Transfer the corn to a plate and let cool slightly. Place the poblanos in a bowl and cover with a plate or plastic wrap for 10 minutes to steam. Peel away as much of the charred skin as possible, using a knife if needed. Remove the stems and scrape out the seeds, then roughly chop.
Cut the kernels from the corn cobs. Set aside ½ cup of the corn and ½ cup of the chopped poblanos for garnish. Reserve the corn cobs to simmer in the stock later.
Meanwhile, heat the butter and olive oil in a Dutch oven or large pot over medium heat. Add the onion and sauté until softened and translucent, about 5-6 minutes.
Add the garlic, smoked paprika, cumin, coriander, and oregano and cook for another minute, until fragrant.
Add the remaining chopped poblanos and corn along with the potatoes and stir to combine.
Pour in the chicken stock, then add the bay leaves and nestle in the reserved corn cobs. Increase the heat to medium-high and bring the soup to a bubble. Reduce the heat to medium-low and simmer, partially covered, for about 15 minutes, or until the potatoes are tender.
Remove and discard the corn cobs and bay leaves. Transfer about ¼ of the soup to a blender and blend until smooth, then pour it back into the pot and stir to combine. Note: The colour of the soup may vary depending on what gets scooped into the blender. More corn and potatoes will give you a yellower broth, while more poblanos will make it greener. Whatever colour you end up with is perfect!
Reduce the heat to low. Add the room-temperature Boursin, breaking it into a few pieces as you add it, and stir until completely melted into the broth.
Stir in the lime juice, then taste and season with salt and black pepper as needed. Add the shredded rotisserie chicken and cook just until warmed through.
Ladle into bowls and garnish with the reserved poblanos and corn, crushed tortilla chips, sliced green onions, lime wedges, sliced avocado, crema and/or cracked black pepper.