Fill a large pot with water and set it over high heat so it can come to a boil.

Set out a large skillet for the spinach and a nonstick skillet for the eggs. Grate the Parmesan, slice the garlic, and cut the lemon into wedges.

When the water comes to a full boil, add the 1 tablespoon kosher salt to the pot.

Add the spaghetti to the boiling water and stir for the first 30 seconds so the strands do not stick together.

Cook the spaghetti until it is just tender, usually 9 to 11 minutes depending on the brand.

When the pasta has about 4 minutes left, scoop 1 cup of the pasta water into a mug or heatproof measuring cup and set it near the stove.

Put the large skillet over medium heat and add 3 tablespoons of the olive oil.

Start the spinach base: Add the sliced garlic to the skillet and cook for 1 to 2 minutes, stirring often, until it smells fragrant and is lightly golden at the edges.

Add the spinach to the skillet a few handfuls at a time. Stir each handful until it starts to collapse before adding more.

Season the spinach in the skillet with a few grinds of black pepper and a small pinch of salt. Cook for 2 to 3 minutes total, until the spinach is fully wilted and any extra moisture in the pan has mostly cooked off.

Pour 1/2 cup of the reserved pasta water into the skillet with the spinach and bring it to a gentle simmer.

Drain the spaghetti in a colander and transfer it right away into the skillet with the spinach.