Cut the corn kernels from the cobs. Combine them in a bowl with the scallions, thyme, ¼ teaspoon salt, and 1/8 teaspoon pepper. Set aside.
In a medium bowl, combine the chili powder, cayenne, oil, and ½ teaspoon salt. Add the shrimp and toss to coat.
Heat grill or grill pan to medium and cook the shrimp, turning once, until cooked through, about 4 minutes.
Meanwhile, sprinkle the cheese over half of each tortilla. Fold them over and grill, turning once, until the cheese melts and the tortillas are crisp, about 2 minutes.
Transfer the quesadillas to a cutting board and cut into wedges.
Divide the shrimp and corn among plates and serve with the quesadillas.