This recipe can be prepared on the grill or stovetop.
If grilling: Preheat grill. While grill heats, toss mushrooms with oil and sprinkle with salt. Cook mushroom in one layer in a grill basket, in batches if necessary. Cook until one side is deep brown, about 6 to 10 minutes; turn and grill other side till a similar color is achieved, another 6 to 10 minutes. Cooking times may vary.
If sautéing: Heat olive oil in large non-stick skillet over medium-high heat. Add a single layer of mushrooms and cook, without stirring, for about 5 minutes or until mushrooms become red-brown on one side. Add salt, flip mushrooms and cook about 5 minutes more, until other side is same color.
While mushrooms cook, assemble quesadillas; distribute half the cheeses and all avocado slices on left half of six tortillas. When mushrooms are done, distribute mushrooms and cilantro leaves, if desired, among tortillas and top with remaining cheese. Fold tortilla in half and grill or warm in skillet until cheese begins to melt, then flip to cook other side. Transfer to cutting board, cut into wedges and serve with salsa verde, tomatoes and additional cheese.