Toss the cucumber and onions together in a medium bowl. Combine the vinegar, sugar and 1/4 cup water in a small saucepan and bring to a boil. Pour the hot vinegar mixture over the vegetables and toss. Chill until ready to serve.
Chop half of the arctic char into rough 1/2-inch chunks. Cut the remainder into a 1/4-inch dice. Put the larger pieces in a food processor and add the mustard and pulse to make a chunky paste. Transfer the paste to a bowl and stir in the rest of the arctic char, the capers, dill, lemon zest and some salt. Form the mixture into four patties, each about 4-inches in diameter.
Heat the oil in a large nonstick skillet over medium heat. Add the patties and cook until golden brown on the bottom, about 1 1/2 minutes. Flip and cook the other side until golden brown, about 1 1/2 minutes.
Put a slice of tomato on the bottoms of each bagel thin and sprinkle with salt. Top each with 1/2 cup of the arugula and a burger. Using a slotted spoon, scoop up some of the pickled cucumbers and onions and mound on top. Cover with bagel tops and serve.
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