In a saucepan over medium heat, soften the onion in 15 ml
(1 tablespoon) oil. Add 500 ml (2 cups) squash and continue cooking for 2 to 3 minutes. Add the broth and bring to a boil. Season with salt and pepper. Cover and simmer gently until the squash is tender, about 15 minutes.
In a blender, purée the cooked squash. Set aside.
In a pot of boiling salted water, cook the spaghetti until al dente. Drain and set aside.
In a large skillet over medium heat, gently sauté the remaining squash in 30 ml (2 tablespoons) oil until tender, about 10 minutes. Add the garlic and tomatoes and continue cooking for about 3 minutes. Season with salt and pepper. add the squash purée and spaghetti, heat through, then add the arugula and cheese. Toss well, and adjust the seasoning.