In a small saucepan, cover the potato with water and bring to a boil. Simmer over moderate heat until tender, about 10 minutes. Drain and pass the potato through a ricer into a bowl; let cool completely. Measure out 1 cup of the mashed potato; reserve the rest for another use.
In a medium bowl, whisk the sour cream with the mayonnaise, lemon juice, garlic and the 1 cup of mashed potato. Whisk in the 2 tablespoons each of dill and parsley and the 1 teaspoon of horseradish; season the potato-dill sauce with salt and pepper.
In a small skillet, heat 1/4 inch of olive oil until shimmering. Add the shallot and fry, stirring, until golden and crisp, 1 to 2 minutes. Using a slotted spoon, transfer the shallot to paper towels to drain; season with salt.
Spread the sauce on the bread slices, arrange the smoked fish on top and garnish with the crispy shallot, grated horseradish and chopped dill and parsley. Drizzle with olive oil and serve.
The potato-dill sauce can be refrigerated overnight.