Prepare a grill for medium heat. Oil the grill grates.
Coat the chicken with the chile powder, granulated garlic, oregano and a pinch of salt.
Meanwhile, combine the tomatoes, scallions, jalapeno, cilantro, lime juice and 1/4 teaspoon salt in a bowl to make a pico de gallo.
Grill the chicken, turning once, until just cooked through, about 6 minutes per side. Transfer to a cutting board and let rest 5 minutes. (Leave the grill on.) Cut into 1/2-inch pieces.
Lay out 3 of the tortillas on a work surface and divide about half of the cheese evenly among them. Top with the chicken, followed by the pico de gallo, and the remaining cheese. Top with the remaining tortillas.
Working in batches if necessary, grill the quesadillas, flipping once, until the cheese is melted and the quesadillas are golden brown, 2 to 3 minutes per side. Transfer to a cutting board and cover loosely with foil while you repeat with the remaining quesadillas. Cut into triangles and serve warm.
You can also cook the chicken and quesadillas over medium heat in a large saute pan, using a small coating of vegetable oil.