Chicken in Chile Sauce (Ají de Gallina)

Ingredients

8 quail eggs (optional)

4 (1/2-inch-thick) slices firm white sandwich bread, crusts discarded

1 cup whole milk

1 medium onion, chopped

2 garlic cloves, chopped

3 1/2 tablespoons vegetable oil, divided

1/2 teaspoon turmeric

3 to 4 tablespoons bottled ají amarillo or ají mirasol paste (sometimes labeled 'crema')

2 cups reduced-sodium chicken broth

1/2 cup grated Parmigiano-Reggiano

1/2 cup walnuts (1 1/2 ounces)

8 small chicken breast halves with skin and bone (5 1/2 pounds total)