Bring a large pot of salted water to a boil. Beat the eggs with a few grinds of pepper in a small bowl; set aside. Cook the bacon in a medium skillet over medium-high heat until browned and crisp, about 7 minutes. Remove the skillet from the heat and stir in 1/2 cup of the boiling water from the pot.
Meanwhile, cook the pasta in the pot of boiling water as the label directs for al dente; add the broccoli in the last 2 minutes of cooking. Reserve 1/2 cup cooking water, then drain the pasta and broccoli and return to the pot over medium-low heat. Add the bacon and stir to coat. Pour in the beaten eggs and gently cook, stirring, until the sauce is thickened and coats the pasta but the eggs are not scrambled, about 2 minutes.
Remove the pot from the heat and stir in the cheese and all but 2 tablespoons of the scallions. Stir in enough of the reserved cooking water to make a creamy sauce. Divide among bowls and sprinkle with the reserved scallions and more cheese.