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Add the tomato paste, mustard, 1 tablespoon of the salt and 2 tablespoons of black pepper.
Let simmer for 20 minutes. Remove from the heat and set aside.
Pour over sliced peaches through a strainer into a nonreactive container and let marinate for 2 hours.
To make the lamb rub mix 1/4 teaspoon of white pepper, 1 tablespoon of black pepper, 2 tablespoons of smoked paprika, 2 tablespoons kosher salt, and remaining 1 tablespoon of brown sugar in a bowl to form the spice rub. Rub lamb with rub and let rest in the refrigerator for 2 hours.
Preheat a grill to high and roast the lamb starting on direct heat until brown on all sides. Turn grill down to a medium temp and move to indirect heat turning the meat constantly for even cooking until an inserted thermometer reads no more than 130 degrees, 45 minutes to 1 hour.
Set aside and let rest for 10 minutes then slice as thin as possible and toss in BBQ sauce.
Serve on the sweet rolls with peaches and arugula.