To prepare sauce, combine the first 5 ingredients.
To prepare dumplings, heat oil in a medium nonstick skillet over medium-high heat. Add leek and 1 tablespoon ginger; sauté 3 minutes. Remove from heat. Combine leek mixture, mirin, salt, black pepper, and shrimp, stirring well.
Working with 1 wonton wrapper at a time (cover remaining wrappers with a damp towel to prevent drying), spoon about 1 tablespoon shrimp mixture into center of each wrapper. Moisten edges of wrapper with water; bring 2 opposite corners to center, pinching points to seal. Bring remaining 2 corners to center, pinching points to seal. Pinch 4 edges together to seal.
Add water to a wok or large skillet to a depth of 1 inch; bring to a boil. Line a bamboo steamer with 15 carrot slices; arrange 15 dumplings on top of carrot slices. Cover with steamer lid. Place steamer in pan; steam dumplings 12 minutes. Remove dumplings from steamer; cover and keep warm. Repeat procedure with the remaining carrot slices and dumplings. Serve with sauce.