Whisk eggs and water in a bowl with salt and pepper until blended. Melt butter in a medium sized skillet over medium low heat and add the egg mixture. Let mixture sit for about 20-30 seconds allowing the edges to start setting, then gently stir with a wooden spoon to form soft curds. Remove from heat just before all the liquid is solidified and allow the residual heat to finish cooking the eggs.
Whisk together the lemon juice, olive oil, salt and pepper. Add arugula and toss.
Slice each bagel in half lengthwise so you have 6 halves. Top each half with 2 slices of cheese, arugula mixture, scrambled eggs and salmon lox all divided evenly between the 6 bagels. Top with sliced caperberries and dill and season with sea salt and pepper. Serve with lemon wedges.