Heat 2 tablespoons butter over medium heat in a large skillet. Cook and stir garlic and onion in the hot oil until tender, 3 to 4 minutes. Stir in wine and lemon juice, increase heat to medium-high, and simmer to reduce by 1/2, about 10 minutes. Reduce heat to low and stir in cold butter, 1 slice at a time. Stir in tomatoes, basil, kosher salt, and white pepper; remove from heat and set aside.

Brush chicken breasts with olive oil and sprinkle with salt and black pepper.

Preheat an outdoor grill for medium-high heat and lightly oil the grate.

Cook chicken on the preheated grill until no longer pink in the center, 15 to 20 minutes. Place equal amounts of goat cheese on top of each chicken breast in the last few minutes of cooking. Spoon sun-dried tomato sauce over chicken.