How to Make It
In a large saucepan, melt the butter in 1/2 tablespoon of vegetable oil over moderately high heat. Add the sliced large onion and season with salt and pepper. Reduce the heat to moderately low and cook, stirring occasionally, until the onion is golden, about 1 hour. Stir in the vinegar and brown sugar and continue cooking until the onion is browned and almost dry, about 10 minutes longer. Remove from the heat and let cool.
Meanwhile, in a medium saucepan, combine the apples with the water and lemon slice and bring to a simmer over high heat. Reduce the heat to moderate and cook, stirring occasionally, until the apples are softened, about 10 minutes. Discard the lemon slice. Transfer the applesauce to a bowl. Stir in the sugar, cinnamon and nutmeg.
In a food processor or on a box grater, coarsely shred the small onion and the potato. Transfer to a colander and squeeze dry. Let stand for 2 minutes, then squeeze again. Transfer the potato mixture to a large bowl. Stir in the flour, matzo meal, egg, 1/4 teaspoon of salt and 1/8 teaspoon of pepper.
In a medium skillet, heat 2 tablespoons of vegetable oil until shimmering. Drop packed teaspoons of the potato mixture into the skillet and flatten them with the back of a spoon. Cook the latkes over moderately high heat until the edges are golden, about 1 1/2 minutes; flip and cook until golden on the bottom, about 1 minute. Drain on paper towels. Repeat with the remaining potato mixture, adding more oil to the skillet as needed.
Arrange the latkes on a platter. Top each latke with caramelized onion, applesauce and sour cream. Serve warm.