Stir together the yogurt, cucumber, lemon juice, olive oil, and dill in a small bowl to combine. Season with salt and pepper, to taste. Cover with plastic wrap and chill in the refrigerator until ready to use.
Place the grated potatoes in a large dish towel and season with salt and pepper. Gather the ends and squeeze to wring out as much liquid as possible.
Heat 1 tablespoon of vegetable oil in a large skillet over medium-high heat. Working in batches, drop spoonfuls of the potatoes into the pan, pressing gently with the back of a spoon to flatten. Cook for 2 to 3 minutes per side, until golden brown. Drain the latkes on paper towels. Continue cooking the latkes, adding more vegetable oil as needed.
To assemble, top each latke with a dollop of yogurt sauce and a slice of smoked salmon. Garnish each latke stack with a sprig of dill.