Assemble Toppings: Combine goat cheese, lemon zest, and garlic in small bowl. Season with salt and pepper to taste. Gently stir in fresh chives. Set aside.
Season the chopped red onion and hard-boiled egg with salt.
Prepare Potato Tartine: Working quickly (as the potato will quickly begin to oxidize), grate the potato (lengthwisinto a large using the large holes of a grater. Squeeze the potatoes over the sink to remove any excess liquid. Season generously with salt and pepper and toss.
Heat clarified butter in a 8-10 inch non-stick skillet over medium-high heat. Once hot, add the grated potato and shape roughly, using a spatula, into a large circle.
Press on the mixture with the back of a spoon to compact it, cover and cook gently for 8-10 minutes or until the bottom is golden brown.
Flip carefully to other side and cook for another 8-10 minutes or until golden brown and crispy.
Remove to cooling rack and allow to cool until barely lukewarm or room temperature.
Assemble Tartine: Once potato cake has cooled, spread the goat cheese mixture on the top. Layer the smoked salmon directly over this and sprinkle with the red onion, hard-boiled egg, and capers. Garnish with freshly chopped chives. Cut into wedges and serve immediately.