In a medium bowl, mix together the softened cream cheese, shredded chicken, pepper jack cheese, soy sauce, garlic powder, and black pepper until well combined.
Place about 1 teaspoon of the filling in the center of each wonton wrapper. Moisten the edges with water, fold to form a triangle, and press to seal tightly.
Heat the vegetable oil in a deep skillet over medium-high heat. Fry the filled wontons in batches until golden brown, about 2-3 minutes per side.
Remove wontons from oil and drain on paper towels. Serve warm with dipping sauce of choice.
Notes
For added flavor, try adding diced green onions to the filling mixture. These wontons can also be baked instead of fried for a healthier alternative.