Stir together mayonnaise, sour cream, chile powder, cumin, and salt in a small bowl. Set aside.
Process Cotija in a food processor until finely ground, about 20 seconds. Stir together Cotija, cilantro, chives, and amchur powder in a small bowl; set aside.
Pull back husks from corn (do not remove husks); remove and discard corn silk. Tie husks of each ear of corn together with kitchen twine to form a handle.
Preheat grill to medium-high (400°F to 450°F). Place corn, husk handles up, in a large stockpot of cold salted water. (Make sure ears of corn are completely submerged.) Soak 10 minutes; drain.
Grill corn until charred on all sides, about 10 minutes, turning occasionally.
Brush corn with mayonnaise mixture, and sprinkle with Cotija mixture. Serve immediately.
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