In a 2- to 3-quart pan over high heat, bring 3 cups water to a boil.
Meanwhile, peel potatoes and cut into 1-inch chunks. Add to boiling water, cover, and simmer over medium heat until potatoes are very tender when pierced, about 15 minutes. Drain potatoes, return to pan, cover, and keep warm.
As potatoes cook, rub thyme over turkey slices and lightly sprinkle with salt and pepper, then coat with flour, shaking off excess.
Place a 10- to 12-inch nonstick frying pan over high heat. When pan is hot, add oil and tilt to coat bottom. Add turkey, without crowding, and brown lightly, turning once, 3 to 4 minutes per batch. Meat should be white in center (cut to test). Transfer turkey to a platter as cooked and keep warm.
Mix broth, sherry, soy sauce, and cornstarch; add to frying pan. Stir over high heat until boiling, about 1 minute. Add salt and pepper to taste.
Mash potatoes and add enough milk to make them creamy. Add salt and pepper to taste.
Place a slice of bread on each plate and top each slice with half the turkey. Mound potatoes equally on meat, pour gravy over the sandwich, and accompany with a dollop of the cranberry sauce.