Mix oil, lemon juice, 1/2 tsp. salt, the spices, garlic, and cilantro together in a medium bowl. Set aside 1/4 cup of the marinade. Add chicken to bowl, turning to coat evenly. Let chicken marinate, chilled, at least 30 minutes and up to 4 hours.
Blend mayonnaise, red peppers, and remaining 1/2 tsp. salt in a food processor until smooth, about 1 minute. Chill until ready to use.
Prepare grill for high heat (450° to 550°). Brush bottom halves of each roll with 1 tbsp. of reserved marinade mixture. Cook chicken, turning once, until grill marks appear and meat is cooked through, 7 to 10 minutes. Grill rolls cut side down until warmed and grill marks appear, about 2 minutes. Let both cool.
To assemble: Set 1 chicken thigh onto bottom half of 1 roll. Spread 2 tbsp. mayo onto cut side of top half. Sprinkle with 1/2 cup arugula and set top of roll on chicken.
Make ahead: Wrap well in plastic wrap and chill overnight. Or make through step 3, then assemble at the last minute.
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.