Have all ingredients and tools (food processor bowl and blade, large metal bowl, rubber spatula) well chilled before you begin. Fill a container large enough to accommodate the bowl with ice.
SEAFOOD FORCEMEAT BASE
Grind the well-chilled fish in the food processor fitted with the metal blade until it gathers into a ball. Blend in the egg, and then slowly add the cream in a thin stream until emulsified. The mixture should be the consistency of soft-serve ice cream. Transfer to the chilled metal bowl, set the bowl in the container of ice, and stir occasionally until it thickens while you prepare the rest of the ingredients. Heat oil to medium in a small skillet. Add green onions and sauté until tender. Remove from heat and let cool. Fold green onions, mustard, tobiko, lobster base, and salt into the puréed mixture with a rubber spatula. Fold in the lobster. Refrigerate for 1 hour. (May be covered and refrigerated up to 2 days on ice.)
Dust a baking sheet lightly with cornstarch.
Set a gyoza wrapper on a work surface, keeping remaining wrappers covered with a dampened cloth while you work. Place about 2 teaspoons of the seafood mixture in the center. Dip a finger in the cornstarch-water mixture and run your wet finger around the edge of the wrapper to make it sticky, fold one side over the other to enclose, and pinch and crimp the edges together to seal. Place on the baking sheet so the dumplings don’t touch. Repeat with the remaining wrappers and filling. Cover the baking sheet with plastic wrap and refrigerate.
Bring water to a low boil in a small saucepan; add the fennel slices and saffron and poach until fennel slices are crisp-tender. Strain off the liquid and spread on a plate to cool.
Bring a large pot of salted water to boil on high heat. Add the dumplings all at once and cook 2 to 3 minutes, or until they are tender but still firm to the touch. Transfer to a heated pasta bowl, with a slotted spoon.
Ladle about 3 tablespoons of the Lobster Butter Sauce into each of 10 heated pasta bowls. Add 5 dumplings to each; sprinkle with diced tomato and the Saffron-Fennel Garnish.
SUGGESTED WINE PAIRING: Frank Family Vineyards Chardonnay (Carneros, California). Creamy with flavors of baked pear and lemon curd.