Preheat oven to 350°. Rinse and pat roast dry. Rub barbecue seasoning over roast. Place in an aluminum foil-lined 13- x 9-inch pan. Pour barbecue sauce and 1 (14-oz.) can chicken broth over roast. Cover tightly with aluminum foil.
Bake at 350° for 1 1/2 hours or until fork tender. Remove roast from pan, reserving drippings in pan. Shred roast with 2 forks.
Carefully pour drippings into a Dutch oven. Add shredded pork, 1 (32-oz.) container chicken broth, and next 4 ingredients. Bring to a boil over medium-high heat; reduce heat to medium-low, and simmer, stirring occasionally, 30 minutes.
Meanwhile, prepare oven fries and sweet potato fries according to package directions; serve with stew.
*2 (5-oz.) cans diced tomatoes may be substituted.
Note: For testing purposes only, we used McCormick Grill Mates Barbecue Seasoning; Alexia Olive Oil, Rosemary, & Garlic Oven Fries; and Alexia Sweet Potato Julienne Fries.
To make ahead: Prepare recipe through Step Store in an airtight container in refrigerator up to 4 days. Reheat in a large Dutch oven while the oven preheats to cook fries.