Combine olive oil, lemon juice and spices to make Chicken Shawarma in a shallow bowl. Add chicken and toss to coat well. Cover and marinate in the refrigerator at least 4 to 8 hours.
Remove from refrigerator. Grill over direct heat, 16 to 20 minutes, turning once, until chicken is at least 165°F.
To prepare Muhammara Sauce, roast red bell peppers over an open burner until blackened. Place in a sealed plastic bag for 10 minutes. Remove stems and seeds and remove blackened skin.
Combine garlic, walnuts and olive oil in a food processor. Pulse until smooth. Add the remaining ingredients and roasted red bell peppers. Pulse until smooth. Refrigerate until ready to serve. Serve Chicken Shawarma with Muhammara Sauce and Tabbouleh.