For the fried shallots: Fill a deep, heavy skillet with 1 inch of oil and heat to 350 degrees F.
Slice the shallots thinly, and separate into rings. Dust with rice flour, and then drop into the hot oil. Fry until golden brown and crispy. Remove to a paper towel to drain. Season with salt and pepper.
For the bearnaise sauce: In a small saucepan over medium-high heat, combine half of the tarragon with the shallots, vinegar and wine. Bring to a simmer and cook until reduced by half, about 10 minutes. Remove from the heat and set aside to cool.
Blend the egg yolks and 1/4 cup of the bearnaise reduction together in a blender. With the blender running, add one-third of the melted butter in a slow, steady stream. Once the butter emulsifies, turn the blender speed up to high and add the remaining butter. Add the remaining half of the tarragon, season with salt and pepper, and give the sauce one more buzz. Set aside in a warm spot to hold.
For the deviled eggs: To make perfect hard-boiled eggs, place the eggs in a saucepan of cold water and bring to a boil over high heat. Once the water boils, immediately turn off the heat and let sit for exactly 14 minutes. Transfer the eggs to a bowl of cold water, let cool, and then peel.
Slice the eggs in half, and then slice a sliver off the bottom so they will stand erect on the plate. Remove the yolks to a food processor.
Season a bowl of cold water very generously with salt, and stir until dissolved. Rinse the egg whites in the salt water to remove any remaining yolk--this way, you season the eggs at the same time.
To the yolks in the food processor, add 1/4 cup of the bearnaise sauce and the creme fraiche, mayonnaise, mustard and lemon juice. Process until completely smooth--you may have to add a few drops of water to loosen. Season with salt.
Transfer the egg yolk mixture to a piping bag fitted with a large circle tip, and pipe the filling into the egg whites. Top with caviar, fried shallots and tarragon tips.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.