Chicken with Peruvian Chile Sauce

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Chicken:

Vegetable oil cooking spray

1 (4 to 5 pound) chicken, cut into 8 pieces

2 tablespoons vegetable oil

Kosher salt and freshly ground black pepper

Sauce:

1 slice whole wheat sandwich bread, crust removed, cut or torn into 1/2-inch pieces

3/4 cup half-and-half

2 tablespoons vegetable oil

1 large or 2 small shallots, sliced

2 cloves garlic, crushed

1 cup low-sodium chicken broth

2 cups (6 ounces) crumbled queso fresco or mild feta cheese

1/2 cup finely grated Parmesan

1/2 cup chopped walnuts, plus 1/3 cup toasted (see Cook's Note)

1 tablespoon Aji Amarillo paste (found in Latin markets or the Hispanic section of grocery stores)

1/2 teaspoon turmeric

4 teaspoons kosher salt

1/2 tablespoon chopped fresh cilantro or flat-leaf parsley leaves