Preheat the oven to 425 degrees F.
Grease a 13x9-inch baking dish.
In a 6-quart stockpot over medium-high heat, melt the butter.
Add the onion, the celery, and the carrots to the butter and cook, while stirring, until tender, about 3-5 minutes.
Add the garlic to the veggie mixture and cook, while stirring, until it is fragrant, about 1 minute.
Stir 1/2 cup of the flour, the salt, and the pepper into the veggie mixture until blended.
Gradually whisk the broth into the veggie mixture.
Bring the veggie mixture to a boil while stirring constantly.
Reduce the heat to medium and cook the veggie mixture until it thickens, about 4-6 minutes.
Add the chicken to the veggie mixture and stir to combine.
Pour the chicken mixture in the prepared baking dish and bake, uncovered, for 20 minutes.
While the casserole is baking, in a large bowl, combine the biscuit mix and the milk until the mix is just moistened.
Using the extra flour, lightly dust a work surface, turn the dough mixture onto the surface, and gently knead it, about eight times.
Roll the dough into a 12x8-inch rectangle.
Sprinkle the dough rectangle with the shredded cheese and the red peppers.
Roll the dough up, jelly-roll-style, starting with a long side and then pinch the seam to seal.
Cut the dough crosswise into 1-inch-thick slices.
Place the dough slices on top of the hot chicken mixture.
Bake the casserole, uncovered, until the biscuits are golden-brown, about 15-20 minutes.
Serve.