Heat olive oil in a large heavy-bottom pot over medium heat. Season the chicken with ½ teaspoon salt and ½ teaspoon pepper and add chicken in an even layer. Cook undisturbed for 5 minutes, then continue cooking while stirring until the chicken is cooked through. Remove from the pot with a slotted spoon and set aside.

To the same pot, add the onion, carrots, celery and remaining salt and pepper. Cook over medium-low heat for 8 to 10 minutes or until the vegetables soften. Add garlic, oregano and nutmeg and cook for about 1 minute, or until fragrant.

Add the chicken broth and milk and bring to a boil. Add the broccoli, bring to a gentle simmer and cover for 20 minutes. Add about 2 cups of the soup to a high speed blender and process until smooth. Take care when blending hot liquids as they will expand as they mix.

Return the pureed soup back to the pot along with the reserved chicken and bring back to a simmer. Add the shredded cheese to a bowl and sprinkle cornstarch on top. Toss to evenly coat the cheese in cornstarch, then add it into the simmering soup one handful at a time. Once all of the cheese has been added, let it simmer for 1 to 2 minutes, or until the cheese has melted and the soup has thickened slightly.

Serve with additional cheddar cheese grated on top, if desired.