Prepare the garlic and olive oil marinade: In a blender or food processor, pulse the garlic cloves with olive oil until the garlic melts into the oil.
Add mustard to marinade: Add the mild mustard to the garlic and olive oil mixture and blend gently until it thickens into a paste.
Prepare vegetables: Cut the onion into large quarters and separate the leaves. Remove seeds from green and red peppers and cut into large pieces. Place all vegetables in a large bowl.
Add vinegar and seasonings to vegetables: Pour white wine vinegar over the vegetables, then add dried oregano, salt, and pepper. Mix well.
Cut chicken and combine with vegetables: Cut chicken breast into bite-sized pieces. Add chicken to the bowl with vegetables and toss to coat. Let sit for 3-4 minutes to allow vinegar to start 'cooking' the chicken and soften it.
Add mustard marinade to chicken and vegetables: Pour the prepared mustard, garlic, and olive oil marinade over the chicken and vegetables. Mix thoroughly to coat.
Marinate: Let the chicken and vegetables marinate for 6 to 12 hours for best flavor, or cook immediately if short on time.
Assemble skewers: Thread the marinated chicken pieces alternately with onion and pepper pieces onto metal skewers.
Cook in air fryer: Preheat air fryer to 200°C (392°F). Place skewers in the air fryer basket and cook for 30 minutes, turning halfway through to brown evenly.
Serve: Remove skewers from air fryer and serve hot with french fries.