Start with a fresh egg. If the egg isn't fresh, the egg white deteriorates which leads to that jellyfish-like tendrils of whites seen with some poached eggs. To test for freshness, drop the egg into water. If it sinks, its fresh and perfect for poaching and souffles. If it stays submerged but lays with its wide-end up, its older but still good for most recipes. If it floats, throw it out.
Pour 1/2 cup vinegar into each of 2 small bowls. Crack 1 egg into each bowl, taking care not to break the yolk. Let stand for 5 minutes.
Meanwhile, bring a medium saucepan of water to a boil over medium-high heat. Using a whisk, vigorously swirl the water until a vortex forms in the center. Slip 1 egg with the vinegar into the vortex and continue to swirl the water with the whisk around the edges of the pan until it returns to a boil. The egg white should wrap tightly around the yolk, forming an oval shape.
As soon as the water returns to a boil, reduce the heat to medium and gently simmer the egg, frequently swirling the water, for 2 minutes.
Using a slotted spoon, lift the egg from water and use kitchen shears to trim any stray pieces of egg white. Place the egg on paper towels and gently blot; transfer the egg to a bowl or plate. Repeat with remaining egg. Season with salt and pepper.