Whisk eggs, 1/2 cup of the cheese, basil and crushed red pepper if using; set aside.
Bring a large pot of salted water to a boil for the pasta.
In a large deep nonstick skillet, saute pancetta and shallots about 3 minutes, add the garlic and saute another minute.
Add the shredded brussels sprouts, and black pepper and saute another 8 -10 minutes, until slightly caramelized. Remove from heat.
Meanwhile cook pasta al dente according to package directions.
Reserve 1 cup of the pasta water just before draining, then drain and add to the skillet and mix well with brussels sprouts.
Slowly, a tablespoon at a time, add 1/4 cup of the water to the egg mixture and mix well, slowly add to pasta stirring well, add additional reserved pasta water 1/4 cup at a time as needed, stirring and finish with the remaining 1/4 cup cheese. Serve right away.