Pat chickpeas dry with paper towels. Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Cook chickpeas in a single layer for 3–4 minutes without stirring, then toss and cook 2 more minutes until golden. Season with salt, pepper, and red pepper flakes. Remove from pan and set aside.
Bring a large pot of salted water to a boil. Cook pasta until just al dente, one minute less than package directions. Before draining, scoop out at least 1 cup of pasta water and set aside. Drain the pasta.
In the same skillet over medium heat, add remaining 1 tablespoon olive oil. Cook garlic for 90 seconds until golden. Pour in cream, stir in Dijon mustard, lemon zest, and nutmeg. Simmer 2–3 minutes until slightly thickened. Lower heat and stir in parmesan until melted and smooth.
Add drained pasta to the skillet and toss to coat. Add 1/4 cup pasta water and keep tossing until sauce clings to the noodles, adding more water as needed. Stir in spinach if using and let it wilt for 1 minute. Add lemon juice and taste for seasoning.
Fold in crisped chickpeas. Divide between bowls and top with extra parmesan, parsley, and red pepper flakes. Serve immediately.
