Preheat oven to 400°F. Toss squash with 1 tablespoon oil on baking sheet. Sprinkle with coarse salt; bake until squash is tender, 20 to 25 minutes.
Beat eggs in large bowl. Add half and half, wine, mustard, and 1 1/2 teaspoons salt; whisk to blend. Add baguette pieces; fold gently into egg mixture. Let soak 20 minutes, stirring occasionally.
Heat 2 tablespoons oil in large pot over medium-high heat. Add onions and sauté until golden brown and caramelized. Set aside. Add 1 tablespoon oil to pot and add kale; cover and cook 2 minutes. Uncover and stir until kale is wilted but still bright green, about 5 minutes (kale will be a bit crunchy). Reduce oven temperature to 350°F.
Generously butter 13x9x2-inch baking dish. Using slotted spoon, transfer half of bread from egg mixture to prepared baking dish. Put half of kale and onion over bread. Put half of squash over bread kale and onion; sprinkle with half of cheese. Repeat with remaining bread, kale, onion, squash, and cheese. Pour remaining egg mixture over bread pudding.Cover bread pudding with foil. Bake 20 minutes. Remove foil; bake uncovered until custard is set and bread feels springy to touch, about 20 minutes longer.
Preheat broiler; broil pudding until cheese browns slightly, about 2 minutes. Cool 5 minutes and serve.