Soak skewers in water for at least an hour before using.
Light grill to medium-low heat. Cut peaches, red onion, and halloumi into roughly the same-sized chunks.Thread, alternating peaches, halloumi, and onion, onto the soaked skewers (roughly 3-4 of each). Continue with remaining skewers. Brush mix with olive oil and sprinkle with salt and pepper.
Grill skewers, turning occasionally, until halloumi is beginning to char, 2-3 minutes per side. Remove from heat.
To make sauce, combine ingredients in a blender or food processor and run until pureed. Serve skewers with basil puree.