1. Preheat grill to 300° to 350° (medium) heat. Cook first 3 ingredients in a 2-qt. heavy saucepan over medium heat, stirring constantly, 2 minutes.
2. Cut each onion crosswise into 3 slices. Grill onion slices, covered with grill lid, 32 minutes, turning every 4 minutes and basting with bourbon mixture with each turn during last half of grill time. Remove from grill, wrap in aluminum foil, and let stand 10 minutes.
3. Meanwhile, whisk together vinegar and next 5 ingredients. Add oil in a slow, steady stream, whisking constantly until smooth. Whisk before serving.
4. Sprinkle tomato slices with salt and pepper to taste. Place 1 tomato slice (or stack 2) on each of 4 plates. Top each with 1 onion slice, and spread with 1 1/2 tsp. goat cheese. Top each stack with 2 basil leaves and 1 bacon piece. Repeat layers once. Top with remaining onions. Spoon 1 Tbsp. vinaigrette over each stack.
5. Toss greens with 3 Tbsp. vinaigrette, and serve immediately with Sweet Onion Stacks and remaining vinaigrette.