Southwestern Cornbread Custard Stuffing

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4 tablespoons unsalted butter, plus more for the dish

2 poblano chile peppers, seeded and chopped

1 red bell pepper, chopped

4 scallions, sliced (white and green parts separated)

1 tablespoon chopped fresh thyme

Kosher salt and freshly ground pepper

6 large eggs

2 1/2 cups low-sodium chicken broth

1 1/2 cups half-and-half

1 15-ounce can creamed corn

6 cups stale 3/4-inch cornbread cubes (about 1 pound)

6 cups stale 3/4-inch brioche cubes (about 12 ounces)

2 1/2 cups shredded Colby jack cheese