Preheat the oven to 400°F (200°C) and line a baking sheet with foil or parchment.
Make the filling: in a mixing bowl, combine the chopped or shredded chicken, 1 1/2 cups salsa, 1 1/2 teaspoons cumin, 1/2 teaspoon dried oregano, 1 1/4 cups shredded cheddar, and chopped green onions; stir until evenly mixed.
Assemble burritos: place a large tortilla on a flat surface, add a heaping 1/2 cup of the chicken mixture in the center, fold the sides in and roll tightly; repeat with remaining tortillas and filling.
Arrange burritos seam-side down on the prepared baking sheet and lightly brush the tops with olive oil.
Bake the burritos at 400°F for 20–25 minutes, or until the tortillas are golden brown and crisp.
Make the green chile sauce: in a medium saucepan over medium heat, melt 3 tablespoons butter. Whisk in 3 tablespoons flour and cook, stirring constantly, about 3 minutes until slightly golden.
Add the minced garlic to the roux and stir 30 seconds, then gradually whisk in 2 cups low-sodium chicken broth until smooth.
Stir in 1/2 teaspoon cumin, 1/2 teaspoon dried oregano, and salt and pepper to taste; cook, stirring, until the sauce thickens, about 2–3 minutes.
Remove the sauce from heat and stir in the 4 ounces diced mild green chilies, 1/3 cup shredded cheddar, and 1/2 cup sour cream until smooth; taste and adjust seasonings if needed.
Broil: remove baked burritos from the oven and switch the oven to high broil. Spoon some green chile sauce over each burrito and sprinkle with additional shredded cheddar.
Broil for 2–3 minutes, watching closely, until the cheese is melted and bubbling. Serve immediately.