In a medium saucepan over medium heat, warm the oil. Add the onion and cook until tender, about 3 minutes. Add the garlic and cook until fragrant, about 1 minute more. Add the remaining sauce ingredients, including ¼ teaspoon salt and ¼ teaspoon pepper, and whisk to combine. Bring the sauce to a boil, and then reduce the heat and simmer gently until thickened slightly and the flavors have blended, about 20 minutes. Remove from the heat. Reserve ¼ cup of the sauce for basting the chicken.
In a medium bowl whisk 3 tablespoons oil, the mayonnaise, mustard, vinegar, sugar, ½ teaspoon salt, and ¼ teaspoon pepper. Add the slaw and basil and turn to coat in the dressing. Let the slaw sit at room temperature while you grill the chicken.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Brush the chicken breasts on both sides with oil and season evenly with ½ teaspoon salt and ½ teaspoon pepper. Brush the cut side of the buns with oil. Grill the chicken, smooth (skin) side down first, over direct medium heat, with the lid closed, until the meat is firm to the touch and opaque all the way to the center, 8 to 12 minutes, turning and basting with the reserved sauce three times. During the last minute of grilling time, toast the buns, cut side down, over direct heat. Remove from the grill and let the chicken rest for 3 to 5 minutes.
Cut the chicken into thin slices and toss with your desired amount of sauce. Fill each hoagie bun with the chicken and slaw. Serve right away.