Combine fresh cilantro, jalapeños, mayonnaise, Cotija or Parmesan, garlic, lime juice, olive oil, salt, and pepper in a blender. Blend until mostly smooth while keeping some texture with green flecks. Taste and adjust seasoning, then chill while you cook.

Slice thick chicken breasts into cutlets for even cooking. Toss with olive oil, cumin, paprika, garlic powder, salt, and pepper. Heat a grill pan over medium-high heat and sear chicken, approximately 5-7 minutes per side, until charred and reaches 165°F (74°C). Let rest for 5 minutes before slicing.

In a saucepan, mix rice, water or broth, butter, and salt. Bring to a boil, reduce heat, cover, and simmer for 15-20 minutes or until liquid is absorbed. Remove from heat and let sit covered for 10 minutes. Fluff with a fork, then fold in chopped cilantro and lime juice.

Fill bowls with a generous scoop of cilantro-lime rice. Layer on perfectly sliced Peruvian-style chicken and drizzle with aji verde. Garnish with fresh cilantro leaves and serve right away for a delightful meal.