Fill saucepan with enough water that waterline sits approximately 1-inch above top of eggs.
Place saucepan on stovetop over high heat and bring water to boil. Once water begins to boil, stir in vinegar and salt. Bring water back to rolling boil.
Working quickly but carefully, use slotted spoon to gently lower 2 to 3 eggs into water, being careful not to crack eggs on bottom of saucepan. Continue until all eggs have been submerged, then start timer. See Notes below for time recommendations.
When timer is finished, quickly but carefully transfer boiled eggs from saucepan directly into large bowl filled with ice bath. Let eggs sit in ice bath at least 5 minutes.
Once eggs are chilled, drain water and remaining ice from bowl. Vigorously shake and toss eggs in bowl to crack shells. Continue until shells feel like mesh and have no large, intact pieces remaining.
Gently peel away and discard egg shells. If needed, hold eggs under running water while peeling, using force of water to remove any small pieces of shell.
Serve peeled boiled eggs immediately, or place eggs in airtight container and refrigerate until ready to serve.