Puree basil, garlic and oil in mini-processor until very smooth. Season with salt and pepper to taste.
Add enough water to medium skillet to measure 1 1/4 inches. Sprinkle salt generously into water. Bring water to simmer over medium heat. Crack eggs 1 at a time and slip into water. Cook until egg whites are just set and egg yolks are still runny, about 3 minutes.
Place hot toast on plates. Top each with Parmesan. Transfer 2 eggs, well drained, to each piece of brioche. Sprinkle egg with salt and pepper. Drizzle with some of pistou and serve.