In a food processor, combine apple cider vinegar, lemon juice, dijon mustard, basil, garlic, and salt. Process until combined. Slowly drizzle in the olive oil until it is emulsified (completely combined) with the rest of the dressing. Stir in the parmesan cheese.
Season the chicken breast with olive oil, oregano, salt, pepper, and garlic powder.
Preheat the grill to high heat. Add the chicken to the grill and immediately lower the heat to medium. Cook for 3-4 minutes on each side or until the chicken reaches 165℉. Let the chicken rest.
In a small pan, add balsamic vinegar and put over medium-low heat. Let simmer until the balsamic vinegar is reduced by half. Once reduced, chop chicken into ½ inch pieces and toss in balsamic glaze.
Combine the lettuce, red peppers, artichoke hearts, red onion, and dressing in a bowl. Separate the mixture into the 4 wraps. Add chicken and feta to each wrap. Depending on the size of the chicken breasts, there may be some leftover chicken.
Roll wraps and wrap with parchment paper. Cut in half and serve.
Notes
Balsamic glaze is easy to make but stir often or the sugar will all stick to the bottom of the pan.
Marinate the chicken for an hour ahead of time if time allows.
Use an instant-read thermometer on the chicken breasts to make sure you remove them at 165℉, so that you don't over cook them.