Turn oven to broil.
Sauté squash, seasoned with salt and pepper, in olive oil on medium heat in a large skillet until browned and tender, about 12 minutes. Remove from pan.
Add sausage to pan, crumbling as you add.
Cook on medium heat until browned, about 5 minutes.
Add garlic and kale and cook for 5 minutes, stirring occasionally.
Add stock, tomato sauce, cooked squash, and bring to a simmer.
Reduce heat to low and cook for 5 minutes to soften kale. Season with salt and pepper.
Add sauce mixture to pasta in a 13×9-inch baking dish.
Top with mozzarella and broil for 3–5 minutes until cheese is bubbly.