Preheat the oven to 350 degrees. Fill a large, deep skillet of water to a depth of 4 inches. Add vinegar and bring to a steady simmer.
Arrange the muffin halves on a baking sheet and, using a round biscuit cutter, cut each half into a round. Cut the cheese slices into 3-inch rounds with the biscuit cutter. Place a slice of cheese on each of the bottom halves. Cut the slice of ham into shreds or ribbons and place on top of the cheese. Transfer to the oven and bake until cheese melts and top halves are toasted, about 4 minutes.
Meanwhile, cut 4 slices from the tomato and spritz each slice with some olive oil. In a separate nonstick saute pan over medium-high heat, saute the tomato slices for 1 minute on each side. Then season with salt, pepper and dried oregano. Set aside.
Break one egg into a small bowl. Hold the edge of the bowl close to the simmering water and gently tilt the bowl to pour the egg into the water. Repeat with the remaining three eggs. Cook until whites are solid and the yolk is still runny, about 3 minutes. Carefully remove the eggs one at a time with a slotted spoon. Transfer to a plate with paper towels to drain.
Remove the toasted muffin halves with cheese and ham from the oven and transfer to a plate. Place a slice of sauteed tomato on top of the ribbons of ham and top with a poached egg. Season with salt and pepper and top with the other half of each muffin. Serve immediately.