Prepare the chicken: Pat the chicken dry. Combine flour, salt, and black pepper in a bowl. Add the chicken and toss until lightly coated.
Brown the chicken: Heat vegetable oil in a large Dutch oven or deep skillet over medium-high heat. Brown the chicken for about 3–4 minutes per side. Remove and set aside.
Cook the vegetables: Add the chopped onion and sliced bell pepper to the same pan. Cook for 3–4 minutes until slightly softened, scraping up the browned bits from the bottom.
Make the sauce: Add peach preserves, barbecue sauce, soy sauce, and water. Stir well and bring the mixture to a gentle simmer.
Simmer the chicken: Return the browned chicken to the pan and coat it with the sauce. Reduce the heat to low, cover, and simmer for 25–30 minutes.
Add water chestnuts: Stir in the drained water chestnuts during the final 5–10 minutes of cooking to help keep them crisp.
Check for doneness: Use an instant-read thermometer to check the thickest piece of chicken. It should reach 165°F (74°C).
Serve: Let the chicken rest for a few minutes, then serve hot over cooked white or jasmine rice with plenty of the peach-barbecue sauce.