1. Whisk yogurt, 1/4 cup oil, 1 Tblspoon garlic, thyme, oregano, salt, pepper, and cayenne in large bowl.
2. Cut chicken into chunks if you are doing kabobs. Stir in chicken (when I don't have time to do kabobs, I just put in whole chicken breasts into marinade).
3. Cover with plastic wrap and refrigerate 3-6 hours.
4. Whisk remaining 1/4 cup oil, remaining teaspoon garlic, basil, and lemon juice in small bowl.
5. Cover and refrigerate while chicken is marinating.
6. Remove chicken from yogurt marinade; discard marinade.
7. Thread each skewer with 2 pieces pepper, 1 piece onion, 2 pieces chicken, and 1 piece onion. Repeat twice more, ending with 2 additional pieces of pepper.
8. Grill over medium heat until vegetables are tender and chicken is no longer pink, turning every few minutes, about 15 minutes total.
9. Transfer to serving platter and brush kebabs all over with lemon dressing.