Make latkes: Put potato and onion in a colander and set in the sink. Let drain, pressing with the back of a spoon occasionally, until no more liquid comes out, at least 30 minutes. Turn potato and onion into a medium bowl and stir in salt, pepper, flour, and egg. Mix well.
Preheat oven to 300°. In a large nonstick frying pan, heat 2 tbsp. oil over medium heat until shimmering. Shape 2 tbsp. potato mixture into a ball and add to pan, flattening slightly with back of a spoon. Shape and flatten 5 more balls the same way and fry until golden brown, about 3 minutes per side. Transfer to a baking sheet and put in oven to keep warm. Repeat with remaining potato mixture, adding more oil to pan as needed.
For gingered yogurt and green apple topping: To same pan used for latkes, add apple, drizzle with vinegar, and toss to coat. Cook until slightly softened, 1 minute. In a serving bowl, mix yogurt with ginger. Spread ginger yogurt on latkes, then top each with a half-slice of apple.
For avocado cream topping: In a food processor, purée avocado, sour cream, salt, pepper, and lemon juice. Dollop avocado cream onto latkes, then top with smoked trout and cilantro leaves.
For salmon roe topping: Spoon sour cream onto latkes, then top with salmon roe and dill sprigs.