Bring a large pot of salted water to a rolling boil, then add the pasta and cook until just al dente, about 8-10 minutes. Drain, saving a cup of pasta water, and set aside.
Meanwhile, heat olive oil in a large skillet over medium heat, until shimmering and fragrant. Add the minced garlic and sauté for about 30 seconds until it becomes aromatic but not browned.
Add the fresh spinach to the skillet in batches, stirring constantly. Cook until wilted and vibrant green, about 2-3 minutes, until the greens release their aroma and become silky.
Pour in the heavy cream and stir gently, bringing it to a simmer. Let it thicken slightly, about 2-3 minutes, until it becomes creamy and coats the spinach evenly.
Stir in the grated Parmesan cheese until melted and smooth. If the sauce seems too thick, add a splash of reserved pasta water to loosen it to a silky consistency.
Add the cooked pasta directly into the skillet, tossing gently to coat each strand with the creamy spinach sauce. Cook together for 1-2 minutes to meld flavors.
Finish by stirring in lemon zest if using, and season with salt and pepper to taste. Give everything a final toss to distribute the seasoning evenly.
Serve immediately, garnished with extra Parmesan if desired. Enjoy the velvety, vibrant green pasta while warm and fresh.