Heat oven to 350°F (convection) or 375°F (conventional).
Shred potatoes with a large whole grater, placing shredded potato into a bowl of cold water to prevent oxidizing.
Drain potatoes and squeeze dry. Spread in a thin layer on one or two sheet pans. (Line with parchment for easier clean-up if desired.) Bake until dry but not brown, stirring once or twice, about 20 minutes. Cool to room temperature.
In a large bowl, whisk eggs, potato starch, garlic and onion powders, salt and pepper. Mix in shredded potatoes, being careful not to mash the potatoes.
Scoop mixture into 24 portions using a 3-ounce scoop, or about 1 heaping 1/4 cup measure. Place on a sheet pan and flatten to about 1/2-inch. Fry immediately or cover and refrigerate up to 4 hours.
To prepare red onions stir together vinegar, water, brown sugar, salt and garlic in a small container until sugar is dissolved. Add sliced onion. (Adjust container size as needed so onions are completely immersed.) Refrigerate at least 4 hours or up to 1 week ahead. Makes about 2 cups.
To serve, heat 1/2-inch oil in a large deep skillet to 350-375°F. Fry potato portions until deep golden brown, about 2-3 minutes per side. Remove to paper towels to drain. (If frying all at once, place a cooling rack over a sheet pan and hold in a warm oven until serving.)
Spoon about 2 tablespoons Labneh on a plate. Arrange two Hash Bae; garnish each Hash Bae with a dollop of Labneh, 1/2 tablespoon salmon roe, a few strips of onion and dill fronds. Serve with spicy ketchup.