Place 2 cups olive oil and the truffle shavings (with liquid) in a bar blender and process until smooth, about 15 seconds. Place the truffle oil in a clean glass container with a fitted lid and refrigerate. Any unused oil may be stored, refrigerated, for up to 3 weeks.
Heat a 3-quart saucepan over medium heat and add 2 tablespoons of the olive oil to the pan. Once the oil is hot, add the onion, parsnip, and celery to the pan. Cook the mirepoix, stirring occasionally, until the vegetables begin to soften, about 3 minutes. Add the garlic to the pan and saute for 30 seconds. Add the white beans, chicken stock, and bay leaf to the pan. Bring the stock to a boil and reduce heat to a simmer. Continue to cook the beans until they are soft, about 2 hours.
Remove the bay leaf from the beans and place the beans in the bowl of a food processor. Puree the beans until smooth. Add the heavy cream through the feed tube and season with salt, to taste, and the black pepper. Drizzle about 4 to 5 teaspoons of the truffle oil into the feed tube and continue to process the beans for another 5 seconds. Transfer the beans to a heatproof bowl and add the chives. Stir to combine, and then cover the bowl to keep beans warm while you prepare the scallops. Set aside.
Season the scallops on both sides with 1 teaspoon salt and 1/2 teaspoon of cracked white pepper. Set a 12-inch saute pan over medium-high heat and add the remaining 2 tablespoons of olive oil to the pan. Once the oil is hot, place the scallops in the pan and saute for 2 minutes on the first side. Add the butter to the pan, placing the cubes in different parts of the pan. Cook the scallops for 1 minute longer, then turn and cook on the second side for 1 minute. Add the lemon juice and tarragon to the pan and cook for 1 minute longer.
To serve the dish, place 1/2 cup of the white bean puree in the center of each of 4 warmed appetizer plates and place 3 scallops on top of each portion of beans. Drizzle the lemon-butter mixture remaining in the pan around the plate. Garnish with more chives and/or tarragon, if desired.