Preheat the oven to 375F; lightly oil an 8 by 8-inch casserole dish with olive oil.
Heat the oil in a large skillet over medium to medium-high heat; add the onion and bell peppers and cook until softened but not browned, about 5 to 7 minutes, stirring occasionally. Add the garlic and cook 1 minute more, stirring constantly.
While the onions and peppers cook, make the sauce. Add the cream cheese and BBQ sauce to a small saucepan over medium-low heat. Once it’s hot, add the chopped pepper jack deli slices and whisk until melted.
Stir together the onion/pepper mixture, brown rice, chicken, and 3/4 cup of the sauce in a large bowl; pour it into the prepared casserole dish, spread it out evenly, and pour the remaining sauce on top.
Cover the dish with foil and bake until bubbling, about 30 minutes (remove the cover during the last 10 minutes).