In a blender or food processor, smoothly purée cilantro, parmesan cheese, oil, ginger, garlic, and lemon juice. Add salt to taste.
Set pizza crust in a 12-inch pizza pan (or on a 12- by 15-in. baking sheet). Spread salsa evenly over crust. Drop cilantro pesto by the spoonful evenly over the pizza. Sprinkle with jack cheese, olives, and jalapeños to taste.
Bake in a 425° oven until cheese is bubbly, about 20 minutes.
Peel, pit, and slice avocado. Lay avocado slices on hot pizza. Cut into wedges.