Preheat oven to 425 degrees F. As soon as the oven is heated, bake Alexia French fries according to package instructions.
Meanwhile, heat a large skillet over medium-high heat. Add chicken to skillet, season with salt and pepper to taste, and cook until no longer pink. Remove chicken from the skillet and set aside.
Reduce the heat to medium. Melt butter in skillet and add onion and bell pepper. Saute until vegetables are tender, approximately 3-4 minutes. Add garlic to skillet and sauté for one more minute.
Add flour to skillet, whisk, and cook for one minute.
Pour chicken broth and milk slowly into the skillet. Whisk constantly to prevent lumps. Add barbecue sauce and increase heat so that the liquid comes to a bubble.
Once the liquid bubbles, turn the heat back down to medium and allow the mixture to simmer until it thickens, approximately 3-4 minutes.
Remove skillet from heat and stir in the cheese and the cooked chicken until all ingredients are combined.
Prepare the sliders by placing open sandwich rolls onto a baking sheet that has been lined with aluminum foil (for easy clean-up). Spoon chicken mixture onto each bun and top each sandwich with a slice of American cheese.
When your French fries have about 5 minutes remaining on their cooking time, place sandwiches into the oven with the fries. Bake sandwiches just until cheese melts, rolls are toasted, and the meat is heated through (approximately 5 minutes).
While sandwiches and French fries are baking, microwave broccoli or other vegetable according to package instructions.
Serve French fries with sliders and veggies!