Bring a large pot of generously salted water to a boil. Cook spaghetti until al dente according to package directions. Reserve 1 cup of pasta water, then drain.
While pasta cooks, heat olive oil in a large skillet over medium-low heat. Add sliced garlic and red pepper flakes (if using). Sauté gently until fragrant and just starting to turn pale gold, about 2 minutes—do not let it brown.
Pour heavy cream and lemon juice into the skillet. Stir and bring to a gentle simmer. Let it bubble lightly for 1–2 minutes until slightly thickened. Season with salt and pepper.
Add chopped spinach to the skillet. Stir just until wilted and vibrant green, about 1 minute. Turn off the heat.
Add drained pasta and Parmesan to the skillet. Toss vigorously with tongs, adding reserved pasta water a little at a time until the sauce is silky and coats each strand. Adjust seasoning.
Divide among bowls. Top with extra Parmesan, lemon zest, a crack of black pepper, and a light drizzle of olive oil. Serve immediately.