Step 1Rinse the fish under cold running water. Open each fish and lay it flat on a cutting board. Cut each in half lengthwise along the spine, using kitchen shears or a very sharp knife to cut through the thick part of the bones.
Step 2Combine the dashi-no-moto, onion tops in a fish poacher or large roasting pan. Heat to simmering. Cover and simmer about 15 minutes to bring out the flavors of the poaching liquid.
Step 3Add the fish. Reduce the heat to low so the poaching liquid is just under a simmer, with no bubbles breaking the surface. Cover and poach until the fish flakes easily when tested with a fork, 5 to 7 minutes. Check the fish closest to the bone for doneness. Remove the fish to a large platter and let cool. Remove the skeleton and any small bones, then cover and chill until ready to serve.
Step 4Cook the soba noodles in a pot of boiling salted water until tender, about 6 minutes. Drain and toss the noodles with the sesame oil. Cool the noodles, then add the sliced green onions and sesame seeds; toss. Cover and chill.
Step 5To make the ginger-soy sauce, combine the soy sauce, lemon juice, sugar, minced green onion, ginger and garlic. Cover and chill.
Step 6To serve, arrange the noodles on a platter and place the chilled fish, skin side down, on top. Sprinkle the crumbled green tea leaves over the top of the fish. Serve the ginger-soy sauce on the side. Garnish with the strips of carrot.