We eat a lot of grilled chicken in the summer. Sometimes it's simply seasoned with salt & pepper, sometimes seasoned with Northwoods seasoning. Sometimes there's a sauce, like this one: Quick Roasted Red Pepper Sauce from March~April 2015 Cook's Illustrated.
It's simple and quick. It's very much like a romesco sauce we once made for with shrimp & grits. The sauce is delicious on the chicken. It's also delicious on top of bread and on top greens as a thick salad "dressing."
Instead of jarred roasted red peppers, we roasted our own on the grill and then guessed at how many to use. We also omitted the garlic and toasted our bread & hazelnuts in the oven. Pulse all that in a food processor with some sherry vinegar, honey, smoked paprika, and cayenne and you have a delicious sauce for atop grilled or sauteed chicken.