Place one-quarter of the salmon chunks, the Dijon mustard, and the mayonnaise in a food processor and process until a smooth paste forms.
Add the remaining salmon, chopped chives, ground coriander, sumac, ground black pepper, and salt to the food processor and pulse 4–5 times until just combined without overprocessing.
Transfer the mixture to a bowl, cover, and chill in the refrigerator for 30 minutes to firm up.
After chilling, shape the salmon mixture into 6 even patties.
Heat the avocado oil in a skillet over medium heat. Cook the patties 4–5 minutes per side, until golden and cooked through.
To assemble, spread tzatziki on both sides of each bun, then add a salmon patty and top with sliced red onion, tomato, and baby arugula.