In a mixing bowl, combine olive oil, ground cumin, paprika, garlic powder, onion powder, salt, black pepper, and lime juice. Whisk until well combined.
Add the chicken breasts to the marinade, ensuring they are fully coated. Cover the bowl and let it marinate in the refrigerator for at least 30 minutes (up to overnight for more flavor).
While the chicken is marinating, prepare the green sauce. In a blender or food processor, combine cilantro, parsley, jalapeño, garlic, mayonnaise, white vinegar, and salt. Blend until smooth.
Gradually add olive oil while blending until you achieve a creamy consistency. Taste and adjust salt if needed.
Preheat the grill or grill pan over medium-high heat. Once hot, remove the chicken from the marinade and discard any leftover marinade.
Grill the chicken for about 6-8 minutes on each side or until fully cooked (internal temperature should reach 165°F).
Remove the chicken from the grill and let it rest for a few minutes. Slice the chicken into strips.
Serve the grilled chicken on a platter, drizzled with the green sauce, and garnish with extra herbs if desired.