Bring a large pot of salted water to a boil and cook the pasta until just shy of al dente, about 8 minutes. Drain, reserving ½ cup of pasta water, and set aside.
While the pasta cooks, heat the olive oil in a large deep skillet over medium heat until shimmering. Add the minced garlic and cook for about 30 seconds until fragrant, stirring constantly to prevent burning.
Add the fresh spinach in batches, stirring until each batch is wilted and vibrant green, about 2 minutes per batch. Remove from heat once all the spinach is wilted but still bright.
Pour in the heavy cream and bring to a gentle simmer, stirring with a whisk until the sauce is slightly thickened, about 2 minutes.
Gradually add the grated Parmesan cheese, whisking continuously until melted and the sauce is smooth and glossy.
Stir in the cooked pasta, tossing to coat evenly with the sauce. If the sauce feels too thick, add a splash of the reserved pasta water to loosen it and make it silky.
Finish by squeezing in the lemon juice and adjusting seasoning with salt and pepper. Taste and add more lemon or seasoning if desired for brightness.
Serve immediately, garnished with extra Parmesan if you like, and enjoy the bright, silky flavors of this simple dish.
