For the slaw: Soak the onions in a small bowl of ice water for 10 minutes to soften the flavor. Meanwhile, heat the olive oil in a nonstick skillet over medium-high heat and cook the pepperoni, stirring, until browned and crispy, about 2 minutes. Transfer to a paper-towel-lined plate to drain using a slotted spoon, reserving the oil in the pan.
Whisk together the vinegar, honey, mustard and salt in a medium bowl. Whisk in the reserved pepperoni oil from the pan. Drain the soaked red onions and pat dry, then fold into the mixture with the cabbage, basil and radicchio. Cover and let marinate for 30 minutes.
For the burgers: Preheat a grill to medium-high heat. Lightly brush the grates with oil. Divide the ground beef into four 6-ounce patties, 4 inches wide and 1-inch thick. Brush each patty with oil and sprinkle both sides with salt and pepper. Grill the burgers until the meat is visibly cooked more than halfway up the bottom, about 4 minutes, then flip and cook to medium rare, about 4 minutes longer. Let rest for 5 minutes.
Place the burgers on the buns and top with the mozzarella. Just before serving, toss the pepperoni with the slaw and divide among the burgers.