Preheat the oven to 400°F and line a baking sheet with foil or parchment paper.
Make the filling by combining the cooked chicken, salsa, 1½ teaspoons ground cumin, ½ teaspoon crushed oregano, 1¼ cups shredded cheddar, and chopped green onions in a mixing bowl; stir until evenly mixed.
Portion about a heaping ½ cup of the chicken mixture into the center of each flour tortilla.
Fold the sides of each tortilla in and roll tightly, placing each burrito seam-side down on the prepared baking sheet; brush lightly with olive oil.
Bake the burritos at 400°F for 20–25 minutes, until the tortillas are golden brown and crispy.
While the burritos bake, make the green chile sauce: melt 3 tablespoons butter in a medium saucepan over medium heat.
Whisk in 3 tablespoons flour and cook, stirring constantly, about 3 minutes until the roux is bubbly and slightly golden.
Add the minced garlic and cook 30 seconds, then gradually whisk in 2 cups low-sodium chicken broth until smooth.
Stir in ½ teaspoon cumin, ½ teaspoon crushed 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 green chilies, 1/3 cup shredded cheddar, and ½ cup sour cream until smooth; taste and adjust seasoning if needed.
Remove the baked burritos from the oven, turn the oven to HIGH broil, ladle a spoonful of the green chile sauce over each burrito, and sprinkle with additional shredded cheddar.
Broil for 2–3 minutes, until the cheese is melted and bubbly; serve immediately.
