- Bring a frying pan half-filled with salted water to a simmer, then add the vinegar. Break each egg into a ramekin and then slip the egg into the simmering water. Using a slotted spoon, gently lift each egg out of the water when it is cooked to your liking (a slightly runny center is good — that takes about 3 minutes). Shake the spoon to drain as much liquid as possible, and gently put each poached egg in a separate small bowl until needed.
- Chop the sun-dried tomatoes with the capers, tarragon, parsley, salt, and pepper to taste, pulsing in a food processor or by hand.
- Using your hands, and working the meat as lightly as possible, combine the meat with the tomato-and-herb mixture and the teaspoon olive oil.
- Shape the meat firmly into 4 patties, each about 1⁄2-inch thick.
- Heat a cast-iron skillet, add the tablespoon olive oil, and wait until it thins and becomes fragrant. Add the burgers to the pan. Cook over high heat for about 2 minutes per side for very rare. Set on a platter of split baguettes or salad greens; top each burger with a poached egg and then a couple of cheese curls. Serve at once.