Cook the spaghetti in plenty of salted boiling water according to the packet instructions. Before draining, save a mugful of the pasta cooking water.
While the pasta cooks, heat the olive oil in a large frying pan. Add the chopped bacon and fry for 5-6 minutes until crispy. Add the garlic if using and cook for 1 minute.
In a bowl, whisk together the whole eggs, egg yolks, most of the grated cheese and plenty of black pepper.
When the pasta is cooked, drain it and add it to the bacon pan. Remove the pan from the heat immediately.
Pour the egg mixture over the hot pasta and toss everything together quickly. The residual heat cooks the eggs into a silky sauce. Add splashes of the reserved pasta water until you get a glossy, coating consistency.
Serve immediately with the remaining cheese and more black pepper on top.