In a large nonstick skillet, heat 1 tablespoon olive oil over medium-high heat. Working in batches, fry the eggs; season with salt and pepper.
Brush the bread with the remaining 2 tablespoons olive oil. Layer the eggs, ham and cheese on half of the bread slices, then set the remaining bread slices on top.
In the same skillet, working in batches, fry the sandwiches over medium heat, turning once, until golden-brown, about 8 minutes.
Toss the carrots and parsley with the vinegar; season with salt and pepper. Serve with the sandwiches.