Prepare kabobs. Whisk together avocado oil, first amounts of salt and black pepper, dried oregano, and first amount of lime juice in a large bowl. Add in chicken, bell peppers, and red onion and toss to coat in marinade. Cover and refrigerate for 30 to 60 minutes to marinate.
Heat grill to medium heat. On metal skewers, skewer marinated chicken with vegetables. Repeat until remainder of vegetables and chicken are used. Discard marinade.
Grill kabobs for 10 minutes, flip, and grill an additional 8 to 10 minutes, or until chicken is golden and cooked through.
While kabobs are on the grill, prepare avocado sauce. Add avocado, second amount of lime juice, soy milk, cilantro, tahini and second amounts of salt and black pepper to a food processor and pulse on high until mixture is smooth and creamy. Remove kabobs from grill and carefully remove skewers. Dollop avocado sauce on top and devour!