In a 4- to 5-quart pan over high heat, bring 2 quarts water to a boil. Add lobster and cook until meat is opaque but still moist-looking in center of thickest part (cut to test), 10 to 12 minutes. Drain lobster; immerse in ice water until cool, 1 to 2 minutes, then drain again.
Cut shell open and pull out meat from lobster tail. Coarsely chop lobster, put in a bowl, and mix with 2 to 3 tablespoons lemon dressing.
Lay a filo pastry rectangle on each of four dinner plates. Arrange half the lobster mixture, tomatoes, then arugula equally on filo. Cover each stack with another rectangle and top equally with remaining lobster, tomatoes, and arugula. Set another pastry, cheese side up, on each stack. Offer more dressing and salt to add to taste.
Filo-parmesan pastry: You will need 6 sheets of filo dough (about 12 by 18 in.), about 3 tablespoons melted butter, and 6 tablespoons grated parmesan cheese. Lay 1 filo sheet flat (cover remaining with plastic wrap to prevent drying) and brush lightly with butter, then sprinkle with 1 tablespoon cheese. Cover with another filo sheet, brush lightly with more butter, and sprinkle with 1 tablespoon cheese. Continue until all the filo is stacked using the last of the butter and cheese on the top layer. Cut into 12 equal rectangles and transfer to two lightly buttered 10- by 15-inch pans, arranging slightly apart. Bake in a 375° regular or conection oven until golden brown, 8 to 12 mnutes. Use warm or cool.