Bring a large pot of salted water to a boil over high heat for the pasta.

Heat the oil and sliced garlic in a large Dutch oven over medium heat. Cook, stirring occasionally, until the garlic slices are golden brown and crisp, about 3 minutes — remove them immediately with a slotted spoon to a small bowl and set aside so they do not become bitter.

Add the onion and a pinch of salt to the oil remaining in the pot. Cook over medium heat until the onion is softened and translucent, about 5 minutes. Stir in the minced garlic and red pepper flakes and cook until the garlic is fragrant and the oil is slightly tinted by the flakes.

Stir in half of the spinach and cook until just wilted to make room in the pot. Add the remaining spinach, broth, and tomatoes, then cover and simmer over medium-low heat until the greens are very tender, about 5 minutes.

Stir in the beans and olives and keep the mixture warm over low heat while the pasta finishes.

Add the spaghetti to the boiling water and cook until al dente. Drain the noodles thoroughly, but do not rinse them.

Add the drained pasta and Parmesan to the Dutch oven with the spinach mixture. Toss over medium-low heat for 1 to 2 minutes until the sauce coats the noodles and the cheese has melted into a light glaze.

Serve immediately, topped with the reserved garlic chips and a final drizzle of extra-virgin olive oil.