Trim stems of the mushrooms. With a spoon, scrape and discard gills from mushroom caps. Brush mushrooms with oil and season with salt and pepper.
To a medium microwave-safe bowl, add the butternut squash and ¼ cup water. Season with salt and pepper. Cover with vented plastic and microwave 8–10 min., until tender. With a fork or potato masher, mash the butternut squash until smooth. Stir in the pesto.
Meanwhile, set a grill or grill pan to high. Add mushrooms and cook 3–4 min. per side, until tender. Add the tortillas to grill and cook 30 sec. per side, until warm.
Let mushrooms cool slightly, then thinly slice. Divided the mixed greens among the tortillas, then top with mushrooms and the cotija cheese. Serve tacos with the butternut squash mash.