Preheat oven to 425 degrees. On a large rimmed baking sheet toss the squash with 2 tablespoons olive oil. Season with salt and pepper and bake for about 30 minutes, tossing once, or until the squash is just tender. Set aside. Reduce the oven temperature to 325 degrees.
While the squash is baking heat 2 tablespoons olive oil in a large skillet. Add the sliced onions and season with salt and pepper. Cook, stirring occasionally, over medium-low heat until golden, about 30 minutes. Transfer the cooked onions to an extra large bowl and set aside. Add the remaining 1 tablespoon olive oil to the same skillet and heat until shimmering. Add the kale and garlic, crushed red pepper and 1 teaspoon of the thyme. Season with salt and cook over medium-high heat, tossing, until the kale is wilted and tender, about 5 minutes. Add the cooked kale to the bowl with the onions. Add the baguette pieces and squash to the bowl with the kale and onions. Set aside.
In a large saucepan melt the butter over medium-low heat. Add the chopped onion and 1 teaspoon thyme and cook until softened, about 5 minutes. Add the flour and whisk until a light golden paste forms, about 5 minutes. Whisk in 1 cup of milk and continuing whisking and cook until very thick – about 10 minutes. Remove from the heat and whisk in the cream, creme fraiche, sugar, 2 teaspoons salt, 1 teaspoon pepper and the remaining 1 1/2 cups of milk. Let the bechamel cool for a few minutes. Whisk in the beaten eggs. Pour the bechamel egg mixture over the squash and kale in the large bowl. Mix well then pour the strata into a lightly greased 13×9 inch baking dish. I also filled another smaller casserole dish. Set the casserole dishes aside and allow them to rest for 30 minutes, pressing down the bread occasionally.
Bake the strata for 1 hour or until almost set. Increase the oven temperature to 475 degrees. Sprinkle the cheese on top of the strata and bake for an additional 10 minutes or until the top is lightly browned. Remove from the oven and leave to rest for 15 minutes before serving.