Bring a large pot of salted water to a boil. Heat a large skillet over medium-high heat and add the olive oil. Add the whole sage leaves and fry until crisp, 30 seconds to 1 minute. Remove with a slotted spoon and drain on paper towels; season with salt.
Add the squash and shallots to the skillet and cook, stirring occasionally, until lightly browned, 3 to 4 minutes. Add the chopped sage and season with 1/2 teaspoon salt and a few grinds of pepper. Add 1 cup of the boiling water and the heavy cream and bring to a simmer. Cook until the squash is tender and the sauce is reduced by half, 8 to 10 minutes.
Meanwhile, add the broken lasagna noodles to the boiling water and cook, stirring frequently, until tender, about 12 minutes. Reserve 1 cup cooking water, then drain. Add the noodles and kale to the skillet with the squash and toss to coat, adding the reserved cooking water as needed to loosen. Remove from the heat, add 1/4 cup grated parmesan and toss.
Divide the noodles among bowls. Top with fried sage leaves and the remaining 1/4 cup parmesan.