Place the eggs in a single layer in a large saucepan. Ensure there's enough space so they don't crack against each other.

Cover the eggs with cold water, ensuring there's at least an inch of water above the eggs to allow for even cooking.

Bring the water to a boil over high heat. Once boiling, reduce the heat to low, cover the saucepan, and let simmer for about 1 minute.

Turn off the heat and leave the saucepan covered. Let the eggs sit in the hot water for about 12 minutes for large eggs (adjust the time slightly if your eggs are smaller or larger).

Transfer the eggs to a bowl of ice water. Chill them for a few minutes until cool enough to handle. This stops the cooking process and makes peeling easier.

Peel the eggs. Gently crack the shell and peel it under running water to help remove the shell cleanly.

Combine the vinegar, water, salt, and sugar substitute in a large saucepan. To infuse the brine with additional flavors, add spices such as peppercorns, garlic cloves, or bay leaves.

Heat the mixture over medium heat, stirring until the salt and sugar substitute are completely dissolved.

Bring the brine to a simmer, then remove from heat. The hot brine will be poured over the peeled eggs to start the pickling process.

Place the peeled, hard-boiled eggs in a large, clean glass jar.

Pour the hot brine over the eggs in the jar, ensuring the eggs are fully submerged. Add more water or vinegar to cover the eggs completely if necessary.

Allow the jar to cool to room temperature, then seal it with a tight-fitting lid.

Refrigerate the pickled eggs. If you let them sit in the brine for at least 24 hours, they will achieve the perfect balance of flavors.