Place a large saucepan or stock pot over medium heat and melt the butter.
Add the diced onions, celery, garlic, and roasted poblanos to the pot. Saute the vegetables until they are tender, stirring occasionally.
Sprinkle in the corn starch and mix well to ensure it absorbs the butter, cooking for about 1 minute to remove the raw taste of the starch.
Slowly pour in the chicken stock while whisking continuously to avoid lumps. Make sure to scrape any bits stuck to the bottom of the pot as the stock warms up.
Allow the soup to simmer gently until it thickens to your desired consistency. If you prefer a thicker soup, create a slurry by mixing additional corn starch with a little water and gradually stir it into the soup until you achieve the desired thickness.
Pour in the heavy cream and let it cook for a couple of minutes to blend the flavors.
Add the shredded Monterrey Jack cheese and chopped cilantro to the pot, stirring until the cheese is fully melted and incorporated into the soup.
Season the soup with salt and pepper to taste, adjusting the seasoning as needed.
Serve the Roasted Poblano and Jack Cheese Soup hot, garnishing with toppings like Pico De Gallo, fresh cilantro, extra Jack cheese, or crispy fried tortilla strips for added taste and texture. Enjoy!