Brothy Beans and Farro with Eggs and Mushrooms

Name: Brothy Beans and Farro with Eggs and Mushrooms
Ingredients: 1 small onion, unpeeled, halved; 1 medium carrot, peeled; 4 garlic cloves, 1 smashed, 3 finely chopped; 1 1/2 cups dried gigante, runner, or lima beans, soaked overnight, drained; 1 cup farro or spelt, soaked overnight, drained; Kosher salt; 2 red Fresno chiles, seeds removed, finely chopped; 1/3 cup plus 3 tablespoons extra-virgin olive oil; 2 tablespoons finely chopped rosemary; 8 ounces maitake or shiitake mushrooms, torn into bite-size pieces; 2 teaspoons Sherry vinegar or red wine vinegar; Freshly ground black pepper; 4 large egg yolks, room temperature