Put a large pan of lightly salted water on to boil. Beat the eggs together in a bowl, season with freshly ground black pepper and stir in most of the cheese.
Put the spaghetti into the pan and cook as per the packet instructions.
Melt the butter in a large frying pan. Using the back of a knife, press down on the garlic bulbs. This helps the flavour to be released. Add to the frying pan and fry for 2 minutes. Remove the garlic cloves with a slotted spoon and discard. The garlic flavour will be left.
Keep the heat on low and when the spaghetti is cooked, scoop out a small cup of pasta water and put to one side then drain the rest and empty the spaghetti into the frying pan. Add the salmon strips and toss it all together and make sure the pasta is well coated in the butter.
The crucial moment. You need a couple of minutes undisturbed. Take the frying pan off the heat and quickly pour in the eggs and cheese. Toss it furiously together and once it begins to thicken, add a splash of the pasta water to loosen it, add more water until you get the desired consistency.
Toss it again and serve immediately with the remainder of the cheese sprinkled on top and some more freshly ground black pepper and freshly chopped chives or parsley.