In a blender, combine fresh cilantro, sliced jalapeños, mayonnaise, Cotija or Parmesan cheese, minced garlic, fresh lime juice, olive oil, salt, and pepper. Blend until mostly smooth but slightly textured—about 30 seconds. Transfer to a bowl and refrigerate for at least 15 minutes.

Slice the chicken breasts into thinner cutlets. In a mixing bowl, toss the chicken with olive oil, cumin, paprika, garlic powder, salt, and pepper until evenly coated. Heat a grill pan over medium-high heat and sear the chicken cutlets for 5–7 minutes on each side until cooked through to 165°F (74°C). Allow to rest briefly before slicing.

In a medium saucepan, combine rice, water or broth, butter, and salt. Bring to a boil over medium heat. Reduce heat to low, cover, and simmer for 15–20 minutes until liquid is absorbed and rice is tender. Remove from heat, let sit covered for 5 minutes, then fluff with a fork, mixing in cilantro and lime juice.

Layer a portion of cilantro-lime rice in each bowl. Slice the chicken cutlets and place them on the rice. Drizzle a good amount of Aji Verde over the chicken and garnish with fresh cilantro leaves.