Bring a large pot of heavily salted water to boil. Cook spaghetti until al dente. Reserve at least 1.5 cups pasta water before draining — this is critical.
While pasta cooks, mix eggs, extra yolk, and grated parmesan in a bowl. Whisk until combined. Season with black pepper. Set aside.
Cook diced bacon in a large skillet until crispy. Turn off heat. Leave bacon and fat in pan.
When pasta is drained, add immediately to the skillet with the bacon off the heat. Toss to coat in the bacon fat.
Add ½ cup of the hot pasta water to the egg and parmesan mixture and whisk quickly to temper it (this prevents scrambling).
Pour egg mixture over the pasta while tossing constantly and vigorously. Add more pasta water splashes while tossing. The heat from the pasta and water cooks the eggs into a creamy, glossy sauce — not scrambled eggs.
Work quickly. The sauce should be silky. Serve immediately.