Toppings: Whisk first 4 ingredients in medium bowl. Let vinaigrette stand 15 minutes at room temperature or refrigerate up to 2 hours.
Prepare barbecue (medium heat). Arrange sausages, peppers and onion on baking sheet. Brush with some of vinaigrette. Sprinkle with salt and pepper. Grill sausages until cooked through and peppers and onion until slightly charred and crisp-tender, turning and basting occasionally, about 12 minutes for sausages and 8 minutes for peppers and onion.
Transfer sausages and vegetables to cutting board. Cut sausages into 1/2-inch pieces and peppers into thin strips.
Final Preparation: Add coals to barbecue, if necessary. Divide dough into 4 equal pieces. Stretch out each piece on floured surface to 9-inch round.
Place 2 dough rounds on grill. Grill over medium heat until top of dough puffs and underside is crisp, about 3 minutes. Turn rounds over. Grill 1 minute. Transfer to baking sheet with well-grilled side up. Repeat with remaining 2 dough rounds. Sprinkle each with 1/4 of mozzarella and Parmesan. Top each with 1/4 of sausage, peppers and onion, then with 1/4 of goat cheese, tomatoes and green onions. Drizzle each with 1-1/2 tablespoons vinaigrette.
Using large metal spatula, return 2 pizzas to grill. Close grill or cover pizzas loosely with foil. Grill until cheeses melt and dough is cooked through and browned, using tongs to rotate pizzas for even cooking, about 5 minutes. Transfer to plates. Repeat grilling for remaining 2 pizzas.