Combine the squash, broth, milk, and garlic in a medium saucepan over medium-high heat. Bring to a boil, then reduce heat to medium-low and allow to simmer for 20 minutes. (Feel free to start the pasta, next step, during this time.) You want the squash very tender after simmering, so pierce it with a fork to make sure it’s quite soft. Set aside.
Preheat oven to 375°F (191°C).
Boil enough water for your pasta (check the package directions). Cook the pasta just until al dente, or when the pasta is still a little firm. Add the kale, if using, and boil for another 1-2 minutes. (If you aren’t using kale, boil the pasta 1-2 minutes past al dente.) Drain the pasta and kale. Set aside.
Pour the warm butternut squash mixture (squash + all the liquid) into a blender or food processor. Add the yogurt, salt, pepper, and nutmeg. Secure the lid and blend until smooth. Pour into a very large bowl and stir in the cheese until combined. It’s ok if the cheese does not fully melt. Taste the sauce, if you’d like more salt/pepper/nutmeg, add more.
Add the pasta and kale to the sauce and stir until combined. It may seem like a lot of liquid, but the pasta will soak it up. Pour everything into an ungreased 9×13-inch baking pan, 12-inch oven safe skillet, or any 3 to 4 quart baking dish. Top with breadcrumbs and an extra sprinkle of cheese, if desired.
Bake, covered with aluminum foil, for 20 minutes. Remove aluminum foil and bake for another 5 minutes, or until bubbly on the sides.
Remove from the oven and serve with an extra sprinkle of cheese and/or a sprinkle of fresh thyme leaves and/or freshly ground black pepper on top.
Cover and store leftovers in the refrigerator for 5 days.
