Cook rice: rinse, then simmer with water and salt for 15-18 minutes. Fluff and set aside.
Roast vegetables: toss bell pepper, zucchini, onion, and broccoli with olive oil, paprika, salt, and pepper. Roast at 425°F for 20-25 minutes.
Make coleslaw: whisk mayonnaise, vinegar, honey, salt, and pepper. Toss with cabbages and carrot. Chill until serving.
Cook chicken: season breasts with salt and pepper. Sear in olive oil for 4-5 minutes per side, brush with BBQ sauce, cook 1-2 minutes more, then slice.
Assemble bowls: divide rice, top with chicken, roasted vegetables, coleslaw, and extra BBQ sauce if desired.
Notes
For extra flavor, brush more BBQ sauce on chicken before serving.