Step 1 Bring a large pot of salted water to a boil and cook the spaghetti until just al dente, about 9 minutes, or according to the instructions on the package.
Step 2 Meanwhile, heat 4 tablespoons of olive oil in a large skillet over high heat. Add the garlic and cook until tender and beginning to caramelize, about 5 minutes, stirring frequently so as not to burn and adjusting the heat if necessary. Remove the garlic from the pan and put it into a large bowl to keep warm near the stove. When the pasta is cooked, add it, still dripping, to the bowl with the garlic and toss to combine.
Step 3Add the remaining 2 tablespoons of olive oil to the pan and turn the heat to high. One at a time, break the eggs into a small bowl and gently pour into the hot skillet, trying to keep them from touching. Cook until the edges are crispy and golden brown, the whites are set and the yolks are still runny.
Step 4Divide the warm pasta among 4 soup plates. Gently slide an egg onto the top of each. Grind plenty of coarse black pepper over each plate and add a generous pinch of sea salt. Serve immediately.