Heat grill to medium. Brush the mushrooms with 1 tablespoon of the oil and season with ½ teaspoon salt and ¼ teaspoon pepper.
Grill, covered, until tender, 8 to 10 minutes per side; thinly slice.
Lightly brush 1 side of each of the tortillas with the remaining tablespoon of oil. Form 4 sandwiches with the tortillas (oiled sides facing out), mushrooms, and cheese. Grill, covered, until the cheese is melted and the tortillas are crisp, 1 to 3 minutes per side.