Boil a large pot of salted water. Cook pasta until just al dente. Reserve 1 cup cooking water, then drain.
Warm olive oil and butter in a wide skillet over medium heat.
Soften the onion in the skillet for 3–4 minutes, stirring occasionally.
Stir in the garlic and cook 30 seconds, just until fragrant.
Add cherry tomatoes and cook until they burst and release juices, 4–5 minutes.
Fold in chickpeas and red pepper flakes. Warm through for 1–2 minutes.
Add spinach and wilt, about 1 minute, tossing gently.
Toss in drained pasta. Splash in reserved water a little at a time to make a glossy sauce.
Season with salt, black pepper, and lemon juice. Stir in Parmesan until creamy.
Finish with fresh basil. Serve hot with extra Parmesan and pepper.