In a large mixing bowl, combine the shredded chicken, taco seasoning, sour cream, shredded cheddar cheese, and chopped green chilies. Stir until all ingredients are well mixed, then set the bowl aside.

In a saucepan over medium-high heat, add the cubed Velveeta and the undrained diced tomatoes with green chilies. Stir occasionally until the cheese is completely melted and the mixture is smooth and creamy.

Lay the tortillas on a flat surface. Spoon 1/2 to 3/4 cup of the chicken mixture into the center of each tortilla. Spread it slightly and roll it up tightly to form a burrito shape.

Place the rolled burritos side by side in a 9×13 casserole dish. Pour the warm queso sauce evenly over the top of the burritos, ensuring they are all covered.

Bake the enchiladas at 350°F for 20–25 minutes. The dish should be heated through and bubbly, and the cheese on top will become wonderfully golden.