Place chicken breasts in a pot; cover with 1 inch water. Bring to a boil. Reduce heat and simmer, covered, until chicken is no longer pink in the center, 10 to 14 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Drain chicken and cool until easily handled, 5 to 10 minutes. Cut into small cubes.
Place chicken cubes in a blender; blend until easily scooped with a tablespoon.
Lay wonton wrappers on a flat work surface. Brush 1 teaspoon tzatziki sauce over the center of each wrapper. Top with 1 teaspoon chicken and 1 teaspoon feta cheese. Fold over and seal edges.
Whisk egg and water together in a small bowl. Brush over top of wontons.
Heat oil in a deep-fryer or large saucepan to 350 degrees F (175 degrees C). Fry wontons in batches until golden brown, about 2 minutes per batch. Drain on paper towels.
We have determined the nutritional value of oil for frying based on a retention value of 10% after cooking. Amount will vary depending on cooking time and temperature, ingredient density, and specific type of oil used.
Per Serving: 84 calories;20.2 g fat;
6.4 g carbohydrates;
4.2 g protein;
16 mg cholesterol;
168 mg sodium.