In a large sauté pan, melt 1 tablespoon of butter over medium-high heat and add 1 tablespoon of oil; when hot, add half of the mushrooms to the pan and cook until golden brown on both sides, about 4 minutes. Transfer the mushrooms from the pan to a paper towel lined baking sheet or a plate. Add another tablespoon of butter and oil to the pan and repeat the process with the remaining mushrooms. Once browned, transfer them to the baking sheet. 

Add the shallot and ham to the pan and cook until the shallot is tender, about 1 minute. Add the mushrooms back to the pan along with the sherry, mushroom broth, heavy cream and thyme. Season with 1 teaspoon salt and ½ teaspoon freshly ground black pepper. Cook until most of the liquid has evaporated, about 5 minutes. Set aside and cover to keep warm. 

Season the scallops on both sides with the remaining 1 teaspoon salt and the ½ teaspoon of white pepper. Heat 2 tablespoons of olive oil in a large sauté pan over high heat. When the pan is hot and just starting to smoke, add half of the scallops. Cook until well-browned, about 2 minutes, and then carefully turn the scallops over. Add 1 tablespoon of butter to the pan and 1 sprig of tarragon. Baste the scallops with the butter for 2 minutes. Transfer the scallops to a baking sheet lined with a paper towel. Wipe out the pan and repeat the process with the remaining scallops. 

To serve the scallops spoon about 1 1/2 tablespoons of the Cauliflower Puree onto the center of each of the four plates. Place 1/3 cup of the mushrooms on top of the puree and top with 3 scallops per plate. Serve immediately.