Dingle Pies (Irish Lamb Pies with Herbs)

Name: Dingle Pies (Irish Lamb Pies with Herbs)
Cook time: 9 h
Total time: 9 h
Cuisine: British
Ingredients: 6 tablespoons unsalted butter (3 ounces); 1 large yellow onion, diced; 2 teaspoons finely chopped fresh thyme; 2 teaspoons finely chopped fresh sage; 2 teaspoons finely chopped fresh rosemary; 1 (2 1/2-pound) boneless lamb shoulder, trimmed and cut into 1-inch pieces; 1/4 cup all-purpose flour; 2 1/2 cups lower-sodium chicken broth; Water, as needed; Fine sea salt, to taste; 4 cups all-purpose flour (about 17 ounces), sifted, plus more for work surface; 1 tablespoon fine sea salt; 1 2/3 cups unsalted Irish butter (such as Kerrygold) (about 11 1/3 ounces), chilled and cut into 1/2-inch pieces; 4 large eggs, divided; 3 to 5 tablespoons ice water