Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Drain, reserving 1 cup of the cooking water.
Meanwhile, heat the olive oil in a separate large pot over medium-high heat. Add the bacon and cook until crisp, 7 to 9 minutes.
Add the pasta, 1/2 cup reserved cooking water and the parmesan to the pot with the bacon. Cook, stirring and adding more cooking water as needed, until coated; season with salt and pepper.
Meanwhile, heat the remaining 2 teaspoons olive oil in a large nonstick skillet over medium heat. Add the eggs and cook until the whites are set but the yolks are still runny, about 4 minutes; season with salt and pepper. Top the pasta with the eggs and chives.