Creamy White Chicken Enchiladas: My Family's Favorite!

Ingredients

• 1 1/2 lbs (680g) boneless, skinless chicken breasts
• 1 teaspoon (5ml) olive oil
• 1 teaspoon (5ml) salt
• 1/2 teaspoon (2.5ml) black pepper
• 1/4 cup (60ml) chicken broth
• 4 tablespoons (57g) unsalted butter
• 4 tablespoons (30g) all-purpose flour
• 3 cups (710ml) milk, whole or 2%
• 1/2 cup (120ml) sour cream
• 1 cup (115g) shredded Monterey Jack cheese, divided
• 1/2 cup (50g) shredded Cheddar cheese, divided
• 1/4 teaspoon (1.25ml) garlic powder
• 1/4 teaspoon (1.25ml) onion powder
• Salt
• pepper
• 10 (6-inch) corn tortillas
• 1/2 cup (120ml) chopped green chiles (canned), drained
• 1/4 cup chopped cilantro for garnish (optional)