Slice the alligator meat across the grain into slices about 1/4 inch thick. Pound them lightly between two sheets of wax paper or inside a large food-storage bag. Cut the slices into pieces about an inch and a half square.
Stir the salt, black pepper and cayenne into the flour. Dust the alligator cutlets very lightly with the flour mixture. Spread the pieces out on a plate and put it into the refrigerator.
Heat the vegetable oil in a saucepan and add the remaining flour mixture. Make a medium-dark roux, stirring constantly with a whisk or wooden spoon until the right color is reached.
Remove the pan from the heat and stir in the onions, bell pepper, and celery. Keep stirring until the vegetables are soft. Then stir in the tomatoes and juice. Return the pan to a burner on the lowest heat.
Heat 2 Tbs. butter in a heavy skillet over medium-high heat. In batches, brown the alligator slices lightly, turning only once. Each side needs less than two minutes to cook thoroughly. Add more butter to the pan if needed for the succeeding batches.
When all the alligator is cooked, lower the heat a little and add the wine and lemon juice. Bring to a boil while whisking. After it bubbles for about a minute, add the contents of the saucepan to the skillet. Add the alligator slices to the skillet, and lower the heat to the lowest setting and allow to simmer. Adjust seasonings to taste.
While that's happening, heat the vegetable oil for frying to about 35o degrees in a medium saucepan. Mix the cornstarch, salt and pepper with 2 Tbs. water in a bowl. Dip the mirliton slices into the cornstarch mixture to coat lightly. Fry in the hot oil until browning just begins--about a minute. Drain the slices of excess oil in a sieve.
Place four to six slices in a star pattern on six dinner plates. Spoon the alligator sauce piquant over the center, with the mirliton slices sticking out. Sprinkle with snipped green onions and serve.