While the pasta is cooking, scramble the egg in a small bowl. Add one of the 4-tbsp portions of Parmesan cheese and stir until combined.
Then heat the bacon fat and butter for the sauce over medium-low heat. Add the minced garlic and cook 3-4 minutes until the garlic becomes crispy and golden. If the pasta is not done yet, turn off the heat. If the pasta is finished cooking, turn the heat down to low and add the pasta to the pan along with some of the reserved pasta water.
Stir to coat the pasta in fat and oil, then add the egg and cheese mixture to the pan. Stir vigorously. A smooth sauce should begin to emerge. If it is too thick, add some more of the reserved water. Then add the cream and turn the heat up to medium. Stir to combine the sauce with the cream; the result should be smooth and slightly off-white. Raise the heat to medium and bring the sauce to a simmer, then add the remaining 4 tbsp of Parmesan cheese little by little. If the sauce becomes too thick, add additional pasta water; if the final product seems thin, add more Parmesan.
Once the sauce has reached the desired consistency, add the chicken and broccoli. Stir to combine everything, then add the pinch of nutmeg (err on the side of less) and stir one last time. Then your pasta is ready to serve! Serve with additional Parmesan cheese and any toppings of your choice.