Whisk eggs in a bowl with salt and pepper then whisk in Parmesan cheese and set aside.
Send trimmed brussels sprouts through a food processor fitted with a slicing blade, thinly slice with a knife, or slice with a mandolin.
Brown chopped bacon in a large skillet, 10 to 12, over medium heat then remove to a paper towel-lined plate to drain, leaving bacon grease in skillet. Add enough extra virgin olive oil to the skillet to equal 2 tablespoons fat total then add shallots and brussels sprouts, and season with salt and pepper. Turn heat up just slightly and sauté until brussels sprouts are tender and caramelized, 8 to 10 minutes. Add minced garlic and sauté for one more minute. Add cooked bacon back into the skillet then stir to combine and remove skillet from heat.
Meanwhile, cook spaghetti in a large pot of salted, boiling water until al dente (add to the boiling water when you start sautéing the brussels sprouts and shallots). Using tongs, transfer the cooked pasta to the brussels sprouts mixture in the skillet then add the egg/Parmesan cheese mixture and start stirring with tongs to evenly coat the pasta, adding a little bit of pasta cooking water at a time to make the sauce smooth. Transfer pasta to plates and serve topped with freshly grated Parmesan cheese, if desired.