Fill a 2-quart saucepan with 11/2 quarts water, salt, potatoes and thyme. Bring to a boil on high heat and cook the potatoes until they are tender, about 15 minutes. Remove and cool. Scoop each potato using a 1/4 teaspoon reserving the potato pulp in a small bowl.
In the bowl with the potato pulp, add the yogurt and parsley and mix with a fork breaking up the potato. Set aside.
In a small skillet on medium heat, cook the shallot, garlic and wine, stirring often, until the mixture is dry but not browned. Add the shallot mixture and the trout to the yogurt and mix gently with a fork to combine. Season with salt and pepper.
Fill each potato with about 1/2 teaspoon of the smoked trout filling. Top each with a morsel of smoked trout. This mixture is equally delicious on hard-boiled eggs.