How to Make It
Stir together mashed potatoes, flour, and egg in a medium bowl until combined. Turn mixture out onto a floured cutting board; form 2 (1/2-inch-thick) logs. Cut logs into 24 pieces; press each piece gently with a fork to create ridges. Chill 10 minutes.
Heat olive oil in a Dutch oven over medium-high. Add onion, carrots, celery, and garlic. Cook, stirring often, until fragrant, 3 to 4 minutes. Sprinkle with oregano, salt, and pepper. Add chicken stock and 2 cups water, and bring to a boil over medium-high. Cook until vegetables are just tender, 5 to 6 minutes. Add green beans and chilled gnocchi, and cook until beans are tender and gnocchi is cooked and rises to top of soup, 3 to 4 minutes.
Heat a nonstick skillet over medium-high. Add prosciutto, and cook, stirring often, until crispy, 3 to 4 minutes.
Divide soup among 4 bowls, and top with crispy prosciutto. Garnish with parsley and radishes.