Place potatoes in large pot. Cover with cold water. Add 1 teaspoon salt and bring to boil over high heat. Boil until potatoes are tender when pierced with skewer, about 8 minutes. Drain. Return to pot. Using potato masher, mash potatoes. Mash in 1/4 cup butter. Stir in 2/3 cup milk. Season to taste with salt and pepper. (Can be made 1 hour ahead. Let stand at room temperature. Before serving, stir over medium heat until heated through, adding more milk by tablespoonfuls if dry.)
Melt 1 tablespoon butter in large skillet over medium-high heat. Sprinkle scallops with salt and pepper. Add scallops; sauté until just opaque in center, about 2 minutes per side. Transfer to plate; tent with foil. Reserve juices in skillet.
Place wine and shallots in small saucepan. Simmer over medium heat until reduced to glaze, about 3 minutes. Stir in cream; simmer 1 minute. Gradually whisk in remaining 1/4 cup butter. Stir in reserved pan juices. Stir in 1 1/2 tablespoons tarragon. Season tarragon sauce to taste with salt and pepper.
Stir remaining 1/4 cup tarragon into warm mashed potatoes. Divide potatoes among 4 shallow bowls. Place scallops atop mashed potatoes. Drizzle with tarragon sauce and serve.