Preheat the oven to 350 degrees F. Soak the lasagna noodles in a bowl of hot water until they begin to soften, about 10 minutes. Meanwhile, heat the olive oil in a large skillet over medium heat. Add the chard stems and garlic and cook until golden, about 4 minutes. Add the chard leaves and cook until wilted, about 3 more minutes.
Mix the ricotta, cream, egg, parmesan, basil, 1/2 teaspoon salt, and pepper to taste in a small bowl.
Brush a 2-quart baking dish with oil and add 3 noodles in a single layer. Top with half of the ricotta mixture, chard and asiago. Repeat with the remaining noodles, ricotta mixture, chard and asiago. Cover with foil and bake until the cheese melts, about 20 minutes. Remove the foil, sprinkle with mozzarella and bake until bubbly and golden, about 5 more minutes.