Cook ham, carrot, leek, and green onions with butter in a large pot over medium heat until vegetables soften, about 5 minutes.
Meanwhile, in a large saucepan, warm vinegar and 1 1/2 in. water over high heat until bubbles form on pan bottom, then reduce heat to medium-low. Break eggs into water, keeping them slightly apart. Cook until whites are firm but yolks are still soft, 3 to 4 minutes. Transfer eggs with a slotted spoon to a plate.
Add broth and 1/4 tsp. pepper to vegetables. Cover and bring to a boil over high heat. Add pasta and boil gently, uncovered and stirring occasionally, until tender, 7 to 8 minutes. Stir in snap peas and return to a boil, then stir in 1/2 cup parmesan, 2 tsp. tarragon, and more pepper to taste.
Ladle into bowls and top each with a poached egg. Serve with more cheese, tarragon, and pepper if you like.