Fill a large saucepan 3/4 full with water and bring to a boil over high heat. Add 1 tbsp white wine vinegar to the water. Reduce heat to medium-low to maintain a gentle boil.
Stir the simmering water to create a whirlpool. Gently crack one egg into the center of the whirlpool. Repeat with the remaining 3 eggs, spacing them apart. Cook for 3 minutes until the egg whites are set but the yolks remain runny.
While the eggs cook, cut each sandwich bun in half. Place the bun halves on a toaster or toaster oven and toast for 2 minutes until warm and lightly golden.
Remove the poached eggs from the water with a slotted spoon and let them drain.
For each burger, place one ham strip on the bottom half of a toasted bun and top with one poached egg.