Bring a large pot of water to a boil over medium heat and salt generously.
Put the bacon in a large high-sided skillet and cook over medium-high heat until crisp, about 10 minutes. Remove from the heat and reserve bacon in the pan.
When the bacon is about halfway cooked, drop the pasta into the boiling water and cook about 4 minutes for fresh or according to the package instructions if using dried.
Meanwhile, whisk the egg and yolks, 1 cup cheese and pepper together in a small bowl.
When the pasta is ready, return the skillet with the bacon to medium heat. Using a ladle, slowly whisk about 1/2 cup pasta cooking water into the egg and cheese mixture until loosened. Reserve some additional cooking water. Drain the pasta and add it to the skillet.
While tossing continually, slowly drizzle the egg mixture over the pasta until it is completely coated. Add more cooking water if pasta seems dry. Add the spinach leaves to the pan and toss until combined. Transfer the carbonara to a serving bowl and serve immediately with more cheese sprinkled over the top.
The key to perfect carbonara is working while everything is piping hot; this assures the egg will cook and produce a silky, creamy sauce that sticks to the pasta.