How to Make It
In a bowl, whisk the mayonnaise, onion soup mix and lemon juice. Cover and refrigerate for at least 30 minutes.
Meanwhile, in a medium bowl, toss the cucumbers with the sugar and 1/2 teaspoon of salt and let stand until pliable, about 20 minutes.
Light a grill or preheat a grill pan. Form the lamb into 4 patties, about 1/2 inch thick. Using your thumb, make an impression in the center of each patty. Season with salt and pepper. Grill over moderate heat, turning once, until nearly cooked through, about 6 minutes total.
Spread the aioli on the bun bottoms and top with the burgers, cucumbers, peppers and arugula. Close and serve.