Preheat the oven to 200°F and set a wire rack inside a baking dish to keep cooked flautas warm.
In a medium bowl, combine 3 cups shredded chicken, 1 cup Monterey Jack cheese, 1/2 cup salsa, 1/4 cup sour cream, the 4-ounce can diced green chiles, and 1 teaspoon chili seasoning; season with kosher salt and black pepper to taste.
Lay one 6-inch tortilla flat, place about 3 tablespoons of the chicken mixture near the edge, then tightly roll the tortilla around the filling and secure the seam with a toothpick; repeat with remaining tortillas and filling.
Pour oil into a large, deep saucepan to a depth of about 3/4 inch and heat over medium until it reaches 375°F (or is hot enough that a small piece of tortilla sizzles immediately).
Working in batches, carefully add 2 to 4 flautas to the hot oil and fry, turning as needed, until evenly golden brown all over, about 2 to 4 minutes per batch; adjust heat to maintain temperature and avoid over-browning.
Transfer fried flautas to a paper towel–lined plate to drain briefly, season immediately with salt, then move them to the prepared wire rack in the warm oven while you fry the remaining flautas.
Remove toothpicks and serve the flautas warm with your desired toppings.