Bring a large pot of salted water to a rolling boil. Add the pasta and cook until al dente, about 8-10 minutes. Drain, reserving a cup of pasta water, and set aside.
While the pasta cooks, heat olive oil in a large skillet over medium heat. Add the finely chopped garlic and sauté until fragrant, about 30 seconds, until it turns golden but not burnt.
Add the fresh spinach to the skillet in batches, stirring constantly until wilted and tender, about 2-3 minutes. If the skillet gets too dry, splash in a bit of the reserved pasta water to help wilt the greens evenly.
Reduce the heat to low and pour in the heavy cream. Stir gently until the cream heats through and begins to thicken, about 2 minutes. This creates a silky, luscious sauce base.
Stir in the freshly grated Parmesan cheese, a handful at a time, until melted and smooth. Squeeze in the lemon juice and season with salt and pepper to taste, adjusting as needed.
Add the cooked pasta to the skillet with the sauce. Toss gently to coat each strand thoroughly, adding a splash of reserved pasta water if the sauce feels too thick or sticky.
Cook for an additional minute over low heat to let flavors meld and the sauce to cling beautifully to the pasta.
Remove from heat, plate the pasta, and finish with a drizzle of olive oil and extra Parmesan if desired. Serve immediately to enjoy the creamy, vibrant dish at its best.