Bean and Walnut Salad with Fried Eggs

Name: Bean and Walnut Salad with Fried Eggs
Ingredients: 1¼ cups dried flageolet beans, soaked overnight, drained; 1 head of garlic, halved crosswise; 1 tablespoon fresh thyme leaves; 1 small dried chile de árbol; Kosher salt and freshly ground pepper; 2 teaspoons Dijon mustard; 2 teaspoons white wine vinegar; 3 tablespoons olive oil, divided, plus more for drizzling; ½ cup chopped toasted walnuts; ½ cup Big Batch of Caramelized Onions (click here for recipe); 1 cup fresh parsley leaves, chopped, plus more leaves for serving; ¼ cup fresh tarragon, chopped; 4 large eggs