In medium non-stick skillet heat oil over medium heat. Add onion; cook 3 minutes. Stir in chopped peach and cook an additional 3 minutes.
Add ketchup, vinegar, sweetener, Worcestershire sauce and garlic powder. Bring to boil; reduce heat to medium-low and simmer for 10 minutes until peach pieces break down.
In blender or food processor, puree sauce until smooth. Reserve half of the mixture in an airtight container and freeze for another use.
Heat grill to medium-high heat; brush grates with oil. Grill chicken; brushing with peach sauce, turning once, until chicken is cooked through and internal temperature reaches 165⁰F (74⁰C) about 20 minutes.
Transfer chicken to a cutting board and let stand 5 minutes before slicing diagonally into thin strips. Discard any remaining BBQ sauce.
Meanwhile, grill peach halves, cut side down, until char marks appear; about 1 to 2 minutes. Cut peach halves into ½-inch slices.
In large bowl, whisk together oil, lemon juice, sweetener, salt, pepper and Dijon mustard until combined.
Add grilled peach slices, arugula, and red onion and toss to coat. Serve with grilled chicken.