Bring 2 quarts water and vinegar to a gentle simmer in a medium saucepan over medium, about 6 minutes. Adjust heat as necessary to maintain a gentle simmer.
Carefully crack eggs into each of 2 small bowls or ramekins, keeping yolks intact. Using a whisk, swirl water in pan in a circular motion until a whirlpool forms. Gently slip 1 egg into center of whirlpool. Using a slotted spoon, swirl water around edge of pan, gently shaping egg as it spins so the egg white forms a tight, neat oval around the yolk. Cook until white is set, about 3 minutes. Using a slotted spoon, remove egg from pan; place on a plate lined with paper towels. Gently blot top of egg with paper towel to dry. Sprinkle with half of salt and half of pepper. Repeat procedure with remaining egg, salt, and pepper.