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Cook the bacon, oil, onion, garlic,1/2 tsp salt and 1/2 tsp black pepper in a large heavy skillet over medium heat until the onion is golden, about 10 minutes.
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Cook the pasta in a large pot of heavily salted boiling water until it is al dente. Meanwhile, whisk together the eggs, Parmesan and 1/4 tsp each salt and pepper in a large bowl. Reserve 1 cup of the pasta cooking water, then drain the pasta.
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Toss the pasta with the egg mixture, onion mixture, arugula and 1/2 cup of the pasta-cooking water until the greens are wilted. Thin the sauce with additional pasta-cooking water, if desired. Season to taste with salt and pepper and serve immediately.