Pour about 2 inches of water into a straight-sided skillet; add vinegar. Bring to a gentle simmer. The liquid temperature will dip as you add eggs; adjust heat as needed to maintain a constant gentle simmer.
Break 1 egg into each of 4 small bowls or ramekins. Gently slip 1 egg into simmering water at the 12 o'clock position. Set a timer for 4 1/2 minutes. After 30 seconds, slip the second egg into simmering water at the 3 o'clock position. After another 30 seconds, slip third egg in at 6 o'clock. After another 30 seconds, put fourth egg at 9 o'clock.
When the timer reaches 1 1/2 minutes left, the first egg should be done (if the white is just set with no translucent areas, the yolk will still be creamy). Gently remove the first egg with a fish spatula or slotted spoon; place on a plate lined with paper towels, and gently blot top dry with a paper towel. Sprinkle egg with a pinch of salt and pepper. At 30-second intervals, working clockwise, repeat procedure with remaining eggs, salt, and pepper.
Drizzle each toast slice with 1 teaspoon oil. Place 2 eggs on each toast slice.