Two days ahead, make the basil oil:
Blanch the basil and parsley in boiling salted water for 20 seconds, and then shock them in ice water. Squeeze the herbs dry in a dishtowel, and then chop them coarsely. Purée the herbs with the canola and olive oils in a blender until the mixture is uniform and bright green, 3 to 4 minutes. Transfer to a bowl, cover, and refrigerate overnight.
Strain the oil through a fine sieve, discarding the solids. Refrigerate overnight. The next day, decant and discard the sediment.
Pickle the papaya:
In a small, nonreactive saucepan, simmer the water, rice vinegar, sugar, salt, spices, ginger, and jalapeño for 2 minutes. Cool the pickling juice; strain into a nonreactive bowl over the papaya. Let the mixture sit, covered in the refrigerator, for at least 1 hour but no longer than overnight. Strain before using the papaya.
Mix the shallot vinaigrette:
In a small bowl, mix the lemon juice, rice vinegar, parsley, salt, and pepper. Slowly whisk in the roasting oil. Mince the roasted shallots and add them to the bowl.
Make the tuiles:
While the peeled potato is warm, pass it through a ricer (a warm potato makes the batter easier to spread). Whip the potato with the butter on medium until smooth, about 1 minute. Mix in the egg whites until incorporated, about 3 minutes. Add the salt, pepper, and herbs.
Make a template by cutting a 2-1/2 -inch round from a plastic yogurt or sour cream lid, slitting an X in the middle so you can cut out the center easily. Leave a 1/4-inch border and cut a small tab handle. Heat the oven to 350°F.
On nonstick sheet pans or regular pans with a nonstick liner, drop a generous teaspoonful of batter in the center of the template. With an offset spatula, spread a very thin layer of batter into the center of the template. Repeat until you have at least 16. It’s a good idea to bake extra in case of burning or breakage (and to nibble on).
Bake the tuiles until golden brown, 8 to 10 minutes, rotating the pan at least once. Start checking after 5 minutes, removing browned tuiles individually as soon as they’re done. Cool on a rack. Store in an airtight container at room temperature.
Pulp the avocado:
In a small bowl, mash the avocado, shallots, and lime juice together with a fork until almost smooth. Fold in the cilantro; season with salt and pepper.
Prepare the lobster:
Tie a spoon or blunt table knife to the lobster tails to keep the tail straight during cooking (a straight tail is easier to slice). Plunge the lobsters into a large pot of boiling salted water. Boil until just cooked, 9 to 10 minutes, shock in ice water, and drain.
To extract the lobster meat, snap off the claws and extract the claw meat. Twist apart the tail and head; discard the head. With scissors, cut along the center of the tail’s underside and then pull the sides away from each other. Extract the tail meat, removing the dark vein if necessary. Set the meat aside unsliced. Just before you assemble the napoleon, slice the meat and toss it with 3 Tbs. of the shallot vinaigrette.
For the final assembly:
Line up all the components on your workspace. Put 1 tsp. pulped avocado in the center of each plate. Center a tuile on top of the pulped avocado, pressing down lightly to secure.
Arrange a few greens on the tuile, and lay a few lobster pieces on top of the greens. On top of the lobster, set a few pieces of pickled papaya.
Top with a teaspoon of pulped avocado and center a tuile on top, pressing lightly to secure. Repeat the layering until you have three layers of lobster and a tuile on top to finish.
Garnish by spooning the remaining shallot vinaigrette around the plate and by drizzling the basil oil around the vinaigrette. Finish with a few grinds of black pepper. Serve immediately.
Make Ahead Tips
The strained basil oil will keep in the refrigerator for up to a month.
nutrition information (per serving):
with 1/2 tsp. basil oil;
Photo: Ben Fink