Combine chicken, olive oil, garlic, red pepper flakes, salt, and black pepper in a large bowl; toss to coat.
Heat a large pot or Dutch oven over medium-high heat. Add chicken mixture and cook, stirring frequently, until chicken is no longer pink in the center, about 5 minutes. Add yellow bell pepper and onion; cook until onion begins to soften, about 5 minutes. Mix in olives and capers until combined. Stir in mozzarella cheese until melted. Remove from heat and stir in feta cheese.
Spread each hoagie roll with 1 tablespoon artichoke aioli. Evenly distribute the chicken mixture among the 4 sandwiches. Sprinkle each sandwich with 1 tablespoon basil.
