Preheat oven to 350ºF and lightly grease a 9 x 13-inch baking dish; set aside.
In a large skillet over medium heat, add oil and when hot, add chicken cubes; season with salt and black pepper. Sauté until no longer pink, about 4 minutes.
Using a slotted spoon, transfer the cubes to a bowl and set aside.
Melt butter in the skillet before adding onions and celery; season with salt. Sauté for 2 minutes.
Add red and poblano peppers; sauté for 2 minutes.
Add garlic and sauté for just 1 minute.
Coat the ingredients with flour and cook for 1:30 minutes. Pour in broth and stir constantly until the mixture thickens and becomes pasty.
Add heavy cream and seasonings; mix well. Add blanched green beans and return chicken with any accumulated juices. Stir to combine.
Pour the mixture into the prepared baking dish. Top with tortilla chips and grated cheese.
Cover with foil lightly greased and transfer to the preheated oven; bake for 15 minutes.
Uncover and bake until the cheese is melted and the top is golden brown, about 10 minutes.
Remove from the heat and let it rest for 10 to 15 minutes before serving.