Pour the water into a wide saucepan and bring to a bare simmer over medium heat (small bubbles clinging to the sides, wisps of steam), 5–8 minutes. Stir in the distilled white vinegar.
Reduce the heat to maintain a gentle simmer; the surface should quiver, not boil.
Set a fine-mesh sieve over a small bowl by the stove. Crack each egg into the sieve to let the thinnest white drain for a few seconds, then tip into the bowl; just before sliding into the water, stir the water to create a gentle vortex if desired and slide in close to the surface. Repeat with the remaining ones, cooking in batches if needed.
Poach 3–4 minutes until the whites are set and the centers are still soft; for slightly firmer yolks, cook 4–5 minutes. Adjust the heat as needed to keep the simmer gentle.
Lift each one out with a slotted spoon and drain on a paper towel for 10–20 seconds; trim ragged edges with the spoon if desired.
Serve immediately and season with kosher salt and black pepper.