In a medium sauce pan, over medium heat, sweat onion in oil, until translucent. Add garlic and saute for 1 additional minute.
Add ketchup, water, lemon juice, brown sugar, mustard, and worcestershire. Whisk to combine. Taste and add a pinch of salt if needed. Add honey, molasses, and hot sauce. Whisk, taste, and add more honey or hot sauce to your preference. For a smokier flavor, add a few drops of liquid smoke (optional).
Once taste is to your liking, thoroughly combine cornstarch (or flour) with a little bit of water, and add to sauce, whisking until smooth. Let sit over medium-low heat until thickened and slowly bubbling.
Meanwhile, shred chicken. Once sauce is thickened, add chicken to pan, and fold in gently (don't stir too hard or you will break up the chicken). Serve on toasted rolls with a slice of red onion.