Combine broth and soy sauce in a medium saucepan and bring to a boil. Reduce heat to barely a simmer.
Mix shrimp, chicken, onion bottoms, garlic, 1 tsp. sesame oil, the ginger, Sriracha, salt, and pepper in a bowl. Arrange egg roll wrappers on a work surface. Shape chicken-shrimp filling into 4 balls and put 1 ball on the center of each wrapper. Moisten wrapper edges with water and pull up all 4 corners of each wrapper to meet in the center. Working from center down, pinch edges together to seal tightly, forming 4-sided pyramids.
Lower dumplings gently into broth, cover, and simmer until cooked through, about 8 minutes. In last 2 minutes of cooking, add shiitakes. Spoon each dumpling into a bowl and ladle broth on top. Sprinkle with onion tops and remaining 1 tsp. sesame oil.
Note: Nutritional analysis is per serving.