Heat 1 tablespoon of the oil in a large skillet over medium. When quite hot, add the mushrooms and cook, stirring, until just tender, 3 to 5 minutes. Add the corn and cook for a minute or two longer, then scrape the mixture into a bowl. Return the skillet to the heat without washing it and add the damp spinach. Cook, stirring constantly, until just wilted, 2 to 3 minutes. Scoop into a colander set in the sink and use a spoon to gently press out excess moisture. Let the vegetables cool, then scoop the spinach in with the mushrooms and corn and season with the salt.
Pour a little of the remaining oil into a small skillet to lightly coat the bottom and set over medium heat until hot. One at a time, quick-fry the tortillas for a few seconds per side just to soften them. As the oil is used up, add a little more, let it heat a moment, then continue frying tortillas. Drain the tortillas in a single layer on paper towels, blotting them dry with additional towels. Cut the tortillas in half.
Heat the oven to 350 degrees. Lightly oil a 13 x 9-inch baking dish. In a medium-size bowl, mix the salsa with the cream or yogurt. Spread a thin layer of the salsa mixture over the bottom of the baking dish, then cover with 8 tortilla halves. Evenly top with half of the vegetables, a scant 11/2 cups of the remaining salsa mixture and 1 cup of the shredded cheese. Add another layer of 8 tortilla halves. Spread with the ricotta and another scant 11/2 cups of the salsa mixture. Top that second layer with a third round of 8 tortilla halves, the remaining vegetables, another portion of salsa mixture and 1 cup of the shredded cheese. For the final layer, evenly lay out the remaining tortillas, spread with the remaining salsa mixture (making sure to completely cover the tortillas) and evenly sprinkle on the remaining 1 cup shredded cheese.
Cover lightly with foil and bake for 25 minutes. Uncover and bake for an additional 10 to 15 minutes, until bubbling and lightly browned. Let stand a few minutes before sprinkling with the cilantro and cutting into squares to serve to your guests on warm plates