Add the egg marinade ingredients to a small pot and whisk to combine. Bring to a boil over medium-high heat, then reduce to low and simmer uncovered for 5 minutes. Remove from the heat, transfer the marinade to a heatproof cup or bowl, and cool completely to room temperature.
Meanwhile, fill a small pot with enough water to cover the eggs by at least 1 inch. Add the salt and white vinegar and bring to a boil over medium-high heat. Gently lower in the room-temperature eggs. If needed, reduce the heat to medium to maintain a controlled boil so the water bubbles steadily without causing the eggs to bounce around and crack. Cook uncovered for 6 1⁄2 minutes for soft yolks, 8 minutes for jammy yolks, or 10 minutes for hard-boiled yolks.
Prepare a large bowl of ice water. Using a slotted spoon, immediately transfer the cooked eggs to the ice bath to stop the cooking. Let them cool completely for 5-10 minutes before peeling.
Gently crack the eggshells with the back of a spoon. Start peeling from the wider end of each egg, where there is an air pocket between the shell and egg white. Place the peeled eggs in a small deep container or mason jar.
Pour the cooled marinade over the eggs, making sure they're submerged as much as possible. If needed, place a square of paper towel over the eggs to help keep the exposed tops coated in marinade while they marinate. Cover and refrigerate for at least 6 hours or 12-24 hours for the best flavor.
Remove as many eggs as desired from the marinade. Slice them in half lengthwise with a sharp knife and enjoy chilled.