Preheat a cast-iron grill pan or skillet over medium-high heat.
Spread 1/4 cup mashed potatoes on to 4 tortillas, leaving 1/2-inch border. Top with a few sprigs of cilantro. Sprinkle 2 tablespoons Cotija and then 2 tablespoons Monterey Jack over the potatoes. Top with the remaining 4 tortillas and brush lightly with some oil.
In two batches, place the quesadillas, oiled-side down, on the grill pan and cook until crisp with grill marks and heated through, about 2 minutes. Brush the tops with a little more oil before flipping and cook an additional 2 minutes. Cut the quesadillas into wedges and top with cilantro leaves. Serve hot with crema and salsa verde if desired.
Put the potatoes in a large saucepan, cover with cold water by about 1 inch and add 2 tablespoons salt. Bring to a boil, uncovered, over high heat, and then reduce the heat to medium-high and simmer briskly until fork tender, about 45 minutes. Drain the potatoes and return them to the pan. Coarsely mash with a traditional potato masher.
While the potatoes are cooking, char the poblanos on all sides under a broiler, 5 to 7 minutes. Place in a bowl and cover with a plate or plastic wrap to cool. Once cool, peel, seed and slice the peppers into thin 2-inch-long strips. Set aside.
Meanwhile, preheat a grill pan over high heat. Lay the onion slices on the pan and grill, turning until cooked through but still firm with distinct grill marks on each side, about 8 minutes total. Let cool and coarsely chop.
Mash the crema into the potatoes and season with salt and pepper. Gently fold in the peppers and onions to make a chunky mashed potato. Serve hot.