Adjust oven rack to be about 6 inches away from broiler and heat broiler. Into a mesh strainer, crack each egg one at a time to strain some of the excess egg white, about 20 to 30 seconds. Pour each egg into individual small bowls.
Fill a Dutch oven or large pot halfway with water, about 6 cups, and bring to a boil. Add vinegar and bring to a simmer with small bubbles barely breaking the surface, about 200 degrees. Hold this temperature.
While water is heating, set a second large pot filled halfway up with water and heat until the water reaches 150 degrees, remove pot from heat and cover to keep warm. The second pot is to keep the poached eggs warm.
Crack eggs into separate ramekins, small bowls, or tea cups. Gently drop one egg into the water at a time, leaving space between them. Cover pot, remove from heat, and let stand until whites closest to the yolk are just set and opaque, about 3 minutes (or 4 minutes for medium-cooked yolks or 6 minutes for hard-cooked yolks).
If the whites are not set after 3 minutes, continue checking every 30 seconds. Using a slotted spoon, carefully lift each egg out and transfer to the pot with the 150-degree water to keep warm.