In a large bowl, beat the softened cream cheese until smooth. Stir in sour cream, 3/4 cup of the cheese, green chiles, garlic powder, cumin, and salt until well combined. Fold in the shredded chicken. The mixture should be thick and creamy, not runny.

Wrap tortillas in a damp paper towel and microwave for 30 seconds until pliable. Working one at a time, spread 2 heaping tablespoons of filling across the lower third of each tortilla. Roll tightly from bottom to top, placing seam-side down on a baking sheet. Do not overfill or they will burst open.

Preheat oven to 425F. Arrange taquitos in a single layer with space between them. Spray or brush lightly with oil. Bake 18-22 minutes, turning once halfway through, until deeply golden and crisp at the edges. Let cool 3 minutes before serving - the filling gets molten hot.