Preheat oven to 400°. Heat 2 tbsp. olive oil in a 2- to 3-qt. pot over medium heat. Add onion and minced garlic; cook, stirring occasionally, until onion is soft and translucent, 5 minutes. Add tomatoes, oregano, and 1/2 tsp. each salt and pepper. Reduce heat and simmer until thick and flavors are combined, about 30 minutes. Set aside.
While sauce is cooking, in a 12- by 15-in. baking pan, sprinkle squash with thyme, remaining olive oil, and salt and pepper to taste. Add garlic cloves and toss squash mixture to coat with oil. Bake until soft, 10 to 15 minutes. Meanwhile, bring 3 qts. salted water to a boil in a large pot.
Reduce oven temperature to 350°. Transfer squash and garlic to a food processor and purée until smooth.
Tear kale leaves from center ribs and discard ribs. Boil leaves until soft, 5 to 8 minutes. Drain; let cool. Squeeze out as much water as possible and chop finely.
In the same pot, bring another 3 qts. salted water to a boil. Add noodles and cook until tender to the bite, about 10 minutes. Drain; rinse with cold water.
In a bowl, mix ricotta, nutmeg, 1 cup mozzarella, and remaining 1/2 tsp. each salt and pepper.
Coat the bottom of a 9- by 13-in. pan with 1/3 of tomato sauce (about 1 1/2 cups). Lay 3 noodles in a single layer over sauce. Top noodles with squash, spreading evenly. Sprinkle 1/2 of kale evenly over squash. Arrange 3 more noodles on kale and top with ricotta, spreading evenly. Top with remaining kale and noodles. Cover noodles with remaining tomato sauce and sprinkle with remaining 1 cup mozzarella.
Bake lasagna until juices are bubbling and cheese is melted, about 30 minutes. Let stand 10 minutes before slicing.
Note: Nutrition analysis is per 4 1/2- by 4 1/2-in. serving.