Preheat grill or grill pan to medium-high heat, approximately 375° to 400° Fahrenheit.
When grill (or grill pan) is hot, transfer chicken from marinade to grill. Grill over indirect heat with lid closed 10 minutes, then flip chicken over and grill another 10 to 12 minutes or until chicken reaches internal temperature of 160° Fahrenheit.
After flipping chicken over, heat large skillet over medium heat. When pan is warm, add butter. Let butter melt completely, swirling pan occasionally to coat bottom with butter.
When butter is melted, add onion and stir, then sauté onion 5 minutes or until translucent.
Pour in remaining ½ cup white wine and 2 tablespoons lemon juice. Stir to incorporate, scraping up any onion that may be stuck to bottom of skillet. Simmer mixture over medium heat 8 to 10 minutes or until liquid has reduced by 50%.
If chicken has not yet been removed from grill, check temperature and transfer chicken to plate once internal temperature reads 160° Fahrenheit. Set aside to rest 5 minutes.
Add fresh basil and sun-dried tomatoes to skillet. Stir to incorporate, then reduce heat to low and keep sauce warm if needed while chicken rests. Stir sauce occasionally.
Transfer chicken to serving plates and top with crumbled goat cheese. Carefully pour or ladle white wine sauce over chicken, making sure to top chicken with sautéed onions and sun-dried tomatoes. Serve dish warm with desired sides.