Fill a small pot with cold water.

Add the eggs to the pot, and turn the heat to medium-high.

Bring the eggs to a rolling boil.

Boil for 10 minutes, then remove from heat. Set aside.

Brown the sausage in a well-greased, 12-inch deep cast iron skillet or Dutch oven. (Add oil or butter to the pot, if needed, to ensure that the sausage doesn't burn.)

Transfer the browned sausage to a paper towel-lined plate.

Begin the Trinity Roux in the pan drippings in the Dutch oven.

While the roux is cooking, put the pot of boiled eggs into the sink. Run cold water over the pot until all of the water in the pot is cold.

Peel the boiled eggs, then slice each one horizontally. Set the sliced eggs aside.

When the roux is ready, reduce the heat to medium-low.

Add the chicken broth, paprika, garlic powder, parsley flakes, hot sauce, and salt, black pepper and cayenne pepper to taste to the mixture. (Be careful with the salt, as the sausage is usually pretty salty).

Cook for 5 minutes, stirring often.

Add the browned sausage, and cook for additional 15 minutes, stirring often. (Add more chicken broth to thin out the gravy, if necessary.)

Add the boiled egg slices in batches to the pot, using a wooden spoon to gently submerge them into the gravy throughout the process.

Taste and adjust seasonings if desired.

Cook for an additional 10 minutes, then remove from heat. (The hot gravy will reduce as it stands, and it may reduce more than you want it to for 4 servings if you do not remove it from the heat immediately. If this happens to you, just add another cup of chicken broth to the gravy and heat it back up for a few minutes to let the broth incorporate into the gravy).

Serve immediately with cooked rice or cauliflower rice.