Add cilantro, jalapeños, mayonnaise, Cotija or Parmesan, garlic, lime juice, olive oil, salt, and pepper to a blender or food processor. Blend until mostly smooth with green flecks, about 30 to 45 seconds. Taste and adjust seasoning. Refrigerate while cooking other components.

If chicken breasts are thick, slice horizontally into thinner cutlets about 3/4 inch thick. In a bowl, toss chicken with olive oil, cumin, paprika, garlic powder, salt, and pepper until well coated.

Heat a 12-inch grill pan or large skillet over medium-high heat. Sear chicken in a single layer, working in batches if needed, for 5 to 7 minutes per side until deeply charred in spots and internal temperature reaches 165°F. Rest 5 minutes, then slice into thick bite-size pieces.

In a medium saucepan, combine rice, water or broth, butter, and salt. Bring to a boil, then immediately reduce heat to low, cover tightly, and simmer 18 minutes until liquid is absorbed. Do not lift the lid during cooking.

Remove rice from heat and rest, covered, 10 minutes. Fluff with a fork, then fold in chopped cilantro and lime juice.

Spoon a generous base of cilantro-lime rice into each bowl. Top with sliced Peruvian-style chicken and drizzle 3 to 4 tablespoons of aji verde over everything. Garnish with extra cilantro leaves and serve immediately.