To make the puree: Melt the butter in a medium saucepan over medium-high heat. Add the onions, salt, and pepper, and cook, stirring, for 3 minutes. Add the shallots and cook, stirring, for 2 minutes. Add the garlic and cook, stirring, until fragrant, about 1 minute. Add the cream and the beans. Bring to a boil, then reduce the heat to medium-low and simmer for 5 minutes. Remove from the heat.

Pour the mixture into the bowl of a food processor or blender. Add additional salt and pepper if necessary, and process on high speed for 30 seconds. With the motor running, add the truffle oil and process until smooth, about 1 minute. Set aside and keep warm.

To make the mushroom sauce: Heat the olive oil in a heavy medium saucepan over medium-high heat. Add the onions, salt, and pepper, and cook, stirring, for 2 minutes. Add the shallots and mushrooms, and cook, stirring, for 2 minutes. Add the garlic and cook until fragrant, about 1 minute. Add the stock and bring to a boil over high heat. Reduce the heat to medium-low and cook until the sauce has thickened, about 4 minutes.

Remove from the heat, cover, and keep warm.

To prepare the scallops: Heat the olive oil in a large heavy skillet on medium-high heat. Season the scallops with the salt and white pepper. Add them to the skillet and sear until lightly golden brown, about 2 minutes on each side.

To serve, spoon about 1/3 cup of white bean puree on each of four plates, place three scallops on the puree, and top with about 1/2 cup of the mushroom sauce.