Combine cooked chicken, salsa, ground cumin, dried oregano, shredded cheddar cheese, and chopped green onions in a large bowl. Mix until well combined.
Place an equal amount of the chicken filling onto the center of each flour tortilla. Fold the sides and roll tightly to form burritos.
Arrange the burritos seam-side down in a lightly greased baking dish and brush tops lightly with olive oil.
In a saucepan over medium heat, melt butter. Add flour and minced garlic, stirring constantly for about one minute to form a roux.
Gradually whisk in chicken broth until smooth. Add ground cumin, oregano, salt, and pepper. Cook for 3 to 4 minutes until sauce thickens slightly.
Stir in diced green chilies, shredded cheddar cheese, and sour cream. Mix until cheese melts and sauce is creamy.
Pour the green chile sauce evenly over the assembled burritos in the baking dish.
Bake in a preheated 375°F (190°C) oven for 20 to 25 minutes, until heated through and bubbly.
Serve hot with optional toppings like extra sour cream, chopped cilantro, avocado slices, or fresh tomatoes.