In a large bowl, combine the chicken legs with lime juice, minced garlic, kosher salt, paprika, black pepper, cumin, oregano, sugar, and olive oil. Mix thoroughly until every piece is evenly coated, then cover and refrigerate for at least 4 hours to allow the flavors to fully penetrate the meat.

Preheat your grill to medium-high heat, about 400–425°F, creating a two-zone fire if possible. Remove the chicken from the marinade and allow excess to drip off before placing on the grill.

Grill the chicken over direct heat, flipping every 3–4 minutes to develop even charring and prevent burning. Cook until the internal temperature reaches 170°F in the thickest part and the skin is crispy and deeply golden, about 18–22 minutes total.

While the chicken cooks, add jalapeños, garlic, cilantro, sour cream, mayonnaise, lime juice, salt, and black pepper to a blender. Blend until smooth, then slowly drizzle in olive oil while blending until the sauce becomes creamy and emulsified.

Transfer the Aji Verde sauce to a bowl and refrigerate until ready to serve. The flavors will intensify slightly as it chills.

Let the grilled chicken rest for 5 minutes before serving. Spoon the Aji Verde generously over the top or serve on the side for dipping, and enjoy immediately.