Whisk all dressing ingredients together until smooth. Set aside.
Melt butter in a skillet over medium high heat. Add bagel pieces and toss to distribute butter. Toast in the skillet, flipping frequently, until bagel pieces begin to turn golden brown. Set aside.
Arrage arugula, cucumber, tomatoes, red onion, eggs, and salmon in 2 large bowls. Top with dressing, bagel pieces, and everything but the bagel seasoning before serving.
You can also toss the arugula and veggies with the dressing before plating for fuss-free serving.