Combine the farro and 1 teaspoon salt in a medium saucepan and cover with water by 2 inches. Bring to a simmer and cook until the farro is tender but chewy, 10 to 20 minutes, or according to the manufacturer's instructions. Drain and set aside.
Heat 2 tablespoons olive oil in a medium pot over medium heat. Add the prosciutto and cook, stirring occasionally, until crisp, about 3 minutes. Remove half the prosciutto with a slotted spoon to a paper-towel-lined plate. Drizzle in the remaining tablespoon olive oil, and add the onion, carrot, celery, garlic, 1/2 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until soft, about 10 minutes. Add the escarole and another 1/2 teaspoon salt and stir for 1 minute to wilt. Pour in the chicken broth and bring to a simmer. Cook for 10 minutes or until the escarole is completely softened. Stir in the cooked farro. Taste for seasoning and add more salt and pepper to taste.
Fill a high-sided skillet with 1 1/2 inches of water. Add the vinegar and 1 tablespoon salt. Bring to a very gentle simmer. Break the eggs, spaced apart, into the water. Cover and turn off the heat. After 3 minutes the whites should be set and the yolks still soft. Remove the poached eggs with a slotted spoon to a paper-towel-lined plate to drain.
Ladle soup into 4 bowls. Top each bowl with a poached egg, a generous sprinkling of Parmesan, the reserved crisped prosciutto and some pepper.
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