Cut the corn kernels from the cobs. Combine them in a bowl with the scallions, thyme, 1/4 teaspoon salt, and 1/8 teaspoon pepper; set aside.
In a medium bowl, combine the chili powder, cayenne, oil, and 1/2 teaspoon salt. Add the shrimp and toss to coat. Heat a 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 in half 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 individual plates and serve with the quesadillas.
Tip: To remove the kernels from a cob, hold the ear by the pointy end and stand the flat end in a large bowl. Cut down the sides with a sharp paring knife. When all the kernels are off, run the edge of the knife down the cob to release any remaining juice.