Heat the oil in a small skillet over medium heat, then add the onions and caramelize, stirring occasionally, until soft and deeply golden brown, 10 to 15 minutes. Deglaze with the cider vinegar and reduce by about half, 3 to 5 minutes. Whisk in the ketchup, soy sauce, brown sugar and paprika, then bring to a boil and let simmer for 5 minutes. Remove from the heat and let cool.
Meanwhile, preheat a grill or a grill pan over medium-high heat. Lightly oil the grill grates, then grill the peaches, flipping once, until they get a nice dark char and are soft, 3 to 4 minutes per side.
Place the cooled onion mixture and grilled peaches in a blender. Puree until smooth.
Sprinkle chicken breasts with salt and pepper. Place chicken on the hot grill and cook for approximately 5 minutes; flip chicken and coat grilled side of chicken with the peach BBQ sauce. Allow chicken to cook for another 5 minutes, then flip again, coating with more peach BBQ sauce.
Cook chicken until it reaches an internal temperature of 165 degrees F, another 4 to 5 minutes on each side.
Meanwhile, toss the sweet potato slices in a bowl with olive oil and salt and pepper to taste. Grill, flipping once, until nicely charred and tender, 3 to 5 minutes per side.
Arrange a few slices of grilled sweet potato on a plate, then top with some Tomato Cucumber Salad and lay a chicken breast on top. Repeat with remaining ingredients. Serve with lime wedges.
Tomato Cucumber Salad:
Combine the tomatoes, cucumbers, onion, avocado oil, ginger, garlic, the spices and lime juice in a large bowl, then season with salt and pepper. Toss well, then toss with the arugula.